Therefore, I pray, let my Lord’s power be great, as You have declared, saying, ‘Hashem! slow to anger and abounding in kindness; forgiving iniquity and transgression; yet not remitting all punishment, but visiting the iniquity of fathers upon children, upon the third and fourth generations.’

Thus will I manifest My greatness and My holiness, and make Myself known in the sight of many nations. And they shall know that I am Hashem.

The Levites Jeshua, Kadmiel, Bani, Hashabniah, Sherebiah, Hodiah, and Pethahiah said, “Rise, bless Hashem your God from eternity to eternity: ‘And they will bless Your glorious name, exalted beyond every blessing and praise!’

Blessed is Hashem, God of Israel, from eternity to eternity.” And all the people said, “Amen” and “Praise to Hashem.”

It was taught in a baraita that Rabbi Yosei said: I was once walking along the road when I entered a ruins among the ruins of Jerusalem in order to pray. Elijah, of blessed memory, showed up.... Elijah said to me, Greetings to you, my Rabbi. I answered him: Greetings to you, my Rabbi, my teacher... And Elijah said to me: What voice did you hear in that ruin? I responded: I heard a Heavenly Voice, cooing like a dove and saying: Woe to the children, due to whose sins I destroyed My house, burned My Temple, and exiled them among the nations. And Elijah said to me: By your life and by your head, not only did that voice cry out in that moment, but it cries out three times each and every day. Moreover, any time that God’s greatness is evoked, such as when Israel enters synagogues and study halls and answers in the kaddish prayer, May His great name be blessed, the Blessed Holy One nods His head and says: Happy is the king who is thus praised in his house. Thus, how great the pain of the Father who exiled his children, and woe to the children who were exiled from their father's table.

...How does the world continue to exist? By the kedusha d'sidra / sanctification that is said in the order of prayers after u'va l'Zion go'el) and by the kaddish response "Yehi Shmei Raba/ let His Great Name" that is recited after the study of aggada.

Rabbi Yehoshua ben Levi said that all who answer "Amen, may His great name be blessed" with all one's might, one's decree of judgment is torn up, as it is stated: “When punishments are annulled in Israel, when the people offer themselves, bless the Lord” (Judges 5:2). What is the reason that punishments are annulled? Because they blessed God. Rabbi Ḥiyya bar Abba said in the name of Rabbi Yoḥanan: Even when one has within him a trace of idolatry, when he answers amen he is forgiven... Reish Lakish said: for one who answers amen with all his strength, the gates of the Garden of Eden are opened, as it is stated: “Open the gates, and a righteous nation shall come who keeps the faith” (Isaiah 26:2). Do not read: Who keeps [shomer] the faith [emunim], but rather: Who say [she’omerim] amen. What is amen? Rabbi Ḥanina said: It is an acronym of the words: God, faithful King [El Melekh ne’eman].

The Gemara asks: Is that to say that one who recites a blessing is preferable to one who answers amen? However, it was taught in a baraita that Rabbi Yosei says: the one who answers amen is greater than the one who recites the blessing.

[Midrash Tana Debei Eliahu]...."yehi shmo..." was translated into Aramaic in Babylonia... it is the foundation of the Kaddish.

May The Tefila (Amida)/ The prayer be recited in any language? Rav Yehuda said: A person should never request that one's needs be met in the Aramaic language, as Rabbi Yoḥanan said, when one prays for his need in Aramaic the ministering angels do not pay attention, since the ministering angels do not recognize/comprehend [makkirin] Aramaic? The Gemara answers: This is not difficult, that (Rabbi Yoḥanan) is referring to the prayer of an individual, who needs the support of the angels, whereas this (the mishna) is referring to communal prayer.

Rabbi Yishmael said, Sasnegir Sar HaPanim conversed with me: my beloved, sit in my lap and I will tell you what will occur to Israel. I sat on his lap and he looked at me and cried, tears flowing from his eyes and falling upon me. I said to him, my splendid radiance, why are your crying? He said to me, come and I will bring you in and let you know what is decreed for Israel, the nation of holy people. He took me by the hand and brought me into the into the room of rooms, the most hidden places and the treasure houses. He took his journal, opened it and showed me writings about the afflictions, each different from the other. I said to him, my splendid radiance, whose are these? And he said to me: Israel’s. I asked him, can Israel withstand them? He said to me, come tomorrow and I will show you miseries different from the first ones. The next day he brought me into the room within rooms and showed me those taken captive, those starving, those pillaged. I said to him, has only Israel sinned? He said to me, every day the harsh decrees are renewed upon them But, since Israel enters the synagogues and houses of study, and answer “amen,t yehei Shemei Raba mevorakh l’olam u’lolmei almaya/ amen, let the Great Name be blessed for ever and throughout eternity" they [the harsh decrees] are not allowed to leave the inner chamber [SLS: i.e., the judgments and decrees are trapped there in potential form and are thus not actualized in the lives of Israel].

Who does not glorify the glorious King, who does not bless the Blessed King, who does not extol the Exalted King, who does not honor the Splendid King, who does not enthrone the King of Kings, who does not praise the Praiseworthy King, who does not sanctify the Holy King, may He be sanctified forever and for eternity? For, every day judgments and wonders are activated before the Holy One, each superior and different than the other, and He delights in us at the time of his children’s prayers. This teaches us how great is the power of the kaddish that prevents afflictions and gladdens the Holy One.

Late Tannaitic - Minor Tractate

This story of Rabbi Akiva's encounter with the soul of a sinner appears in many versions including the one below (not exact translation of the Hebrew text).

"The French halakhic work Mahzor Vitry (12th century) via this story, links this notion, (i.e., that the living can do good for the dead) with the Kaddish." - Alyssa Gray, My People's Prayer Book, Vol.6, p 157.

[Also see "Redeeming Powers Ascribed to the Ḳaddish", Jewish Encyclopedia

http://www.jewishencyclopedia.com/articles/9110-kaddish#anchor2]

See codification of this below - ReMA on Shulhan Arukh Yoreh Deah 376:4.

Rabbi Akiva once saw a man struggling with a heavy burden on his shoulders and bemoaning his lot in life. Rabbi Akiva asked the man what his story was. The oppressed laborer replied that he was the soul of a person who committed every conceivable sin and that if he stopped to talk, he’d get in even more trouble.

The punishment of this particular sinner was to gather wood, which was used to burn him every day. Rabbi Akiva asked if there was any way to free this soul and the deceased replied that the only way was if he had a son who would stand in front of the congregation and say “Barchu es Hashem hamevorah” or “Yisgadal v’yiskadash…,” after which the congregation would reply, “Baruch Hashem hamevorah l’olam voed” or “Yehei shmei rabbah…,” respectively. (These are the prayers of Barchu and Kaddish, in which the leader of the service calls upon the congregation to praise God, which they then do.)

Finally, Rabbi Akiva asked the man who had survived him; the spirit replied that his wife had been pregnant when he died. Rabbi Akiva recorded the name of the deceased, the man’s wife, and his hometown so that he might investigate the matter.

Hurrying to the man’s city, Rabbi Akiva discovered that the deceased was particularly reviled by the townspeople. He had been a corrupt tax collector who took bribes from the rich and oppressed the poor. Among his more notorious deeds, the man had violated a betrothed girl on Yom Kippur! Rabbi Akiva located the widow, who had given birth to a son. So despised was her husband that no one had even circumcised the child. Rabbi Akiva took care of this grievous omission and, when the child was old enough, he taught him Torah and how to daven in shul, including the prayers the man had specified.

As soon as the boy recited the appropriate prayers, his father’s soul was relieved of its harsh punishments. The man’s spirit re-appeared to Rabbi Akiva in a dream to thank the scholar for saving him from the tortures of Gehinnom.

​https://www.ou.org/torah/tefillah/kaddish/why_do_we_say_kaddish_for_the_deceased

Medieval Commentators: 12th-15th cent.

Thus, one says Kadish in Aramaic. Since the prayer is so fitting and such great praise, it was established in the language of the Targum [the vernacular], so that the angels would not understand and become jealous. It is not that there are not other beautiful prayers in Hebrew, but is its apparent, as it is said in Sota 49a that the world only continues to exist because of the Kedusha d'Sidra and the response Yehei Shmei Rabba after agada. The people were accustomed to say Kaddish after a derasha, and, being ordinary folk, they did not understand the Holy [Hebrew] Tongue at all, so it was fixed in the vernacular which everyone understood.

Arba'ah Turim Yaakov ben Asher

Cologne 1270–Toledo c.1340

atYitgadal v'yitkadash is based on the verse v'hitgadalti...that was said about the war of Gog and Magog when the Name of the Holy One will become great, as it is said in Zekharia "on that day Hashem will be one and His Name one"...

Why do we say Kadish in the language of the Targum (Aramaic)? Some say it is so that the angels won't be jealous of us when we offer such substantial praise. Also, because it reminds the Holy one of the destruction of the Temple and the exile of Israel...When Israel answers amen y-sh"r, The Holy Blessed One nods; for a Father to be praised by his children and to have distanced them from his table, He will, as it were, worry about them. And, if the Angels understood, they would testify against us. Thus we say it [Kadish] in the language they do not understand, because they do not recognize Aramaic. Rabbi Yosi explained the reason, as cited in Sotah 49b, that the people were used to saying Kadish after a derasha... and they were ordinary people who only recognized Aramaic, their vernacular, so they used to say it in the language that everyone understood.... [so that] one's decrees of judgment will be torn up (as described in Shabbat 119b)....

[See source #15 above for] Siddur of Rav Amram Gaon [which is quoted here]...

And why was the use of the word nehemata accepted? It was written that the Holy One was bothered by our exile and thus we pray that He will soon and quickly be consoled and redeem us. Thus is it was the custom to say "remember your compassion and love (Ps. 25:6)" when the sh"tz (prayer leader) would begin the kaddish to arouse the Trait of Compassion [so God would] remember us in that moment - and, similarly, one would say the verse "let Your power be great..." that uses the words similar to yitgadal said by the sh"tz.

Yosef Karo 1488-1575

One should have concentration when answering to kaddish. One should answer loudly and run to hear kaddish.

HaMapa (Moses Isserles 152-1572- the Rema's gloss): Answer with the congregation even if you arrive in synagogue after the leader already said yitgadal...and the prayer leader also says yhei Shmei Raba. And when you begin Yitgadal, intend the words of Numbers 14:17-18.

The ReMA's gloss explaining Askenazi minhag: And when the son prays and sanctifies (says Kaddish) in public, he redeems his father and mother from Geihenom.

Rabbi Yechiel Michel Epstein (1829–1908)

The Kaddish is great and awesome praise that the Men of the Great Assembly fixed after the destruction of the First Temple. It is a prayer about the desecration of the Blessed Name from the destruction of the Sanctuary, the decimation of the Holy Land, and the dispersion of Israel to the 4 corners of the earth. We pray 'let the Blessed name be magnified and sanctified' as the prophet said: "I will make myself greater and more sanctified and be known to the nations, and they will know that I am Hashem." As R. Yosi said at the beginning of Berakhot, once a Bat Kol was heard cooing like a dove and saying: 'Oy to the children, that because of their sins, I destroyed My house, and burned my temple, and exiled them...' And Eliahu the prophet said to him, that three times each day a Bat Kol like this is heard. And further, at the time Israel enters its synagogues and study halls, and they answer 'amen! yehei shmei raba...amen ! May His great Name be blessed', the Holy One of Blessing nods his head and says, happy is the King who is praised like this in His house. What it is to be a Father who exiled His children? And woe to the children who were exiled from their Father's table.' And because of its elevated status, it was decreed that it [the Kaddish] would be recited in Aramaic, because in Babylonia they spoke this language; and, thus, so that everyone would understand, it was established in the vernacular. And, additionally, there are hidden reasons for this.

The Sages said in Shabbat 119b that all who answer "amen yehei shmei raba" with all their strength, have the decrees of their judgements torn up, and the gates of the Garden of Eden are opened for them. And in the Pesikta a story is told of Rabbi Yishmael ben Elisha who said: when Israel enter the synagogues and say "amen yehei shmei raba" in a loud voice, the harsh judgments are nullified (see Tosafot there). And toward the end of Sotah it is written that "amen! yehei shmei raba" is one of the foundations of the world. And thus one must be very meticulous in this, and answer with intention and in a loud voice.

And one answers "amen" after every brakha/ blessing. It is an obligation for everyone who hears a blessing to answer "amen", whether or nor not they fulfull a halakhic obligation. Not answering "amen" is considered a great transgressinon.

Amen has 3 intentions:

(1) Halevai - "If only!" as in "amen, thus should God say!"

(2) The strengthening of "Emuna/faith" - for all the blessings of the Amida, such as with "enlivens the dead" we intend that we come to believe in the revival of the dead.

(3) Affirmation of "emet/truth" - for example, it is true that You are the shield of the ancestors, or that You are the Holy Divinity, and similarly for all of the blessings over pleasures and mitzvot, and the middle blessings (of the Tefila), or having full faith that You are the Redeemer of Israel, and all the Morning Blessings, and verses of praise and blessings surrounding the Sh'ma - all are affirmations of Truth.

But with kaddishes, we must intend the first explanation: Let it be Your will that the Blessed Name will be expanded, and "let Your great name be blessed.

And, the essence of faith depends on "amen." As soon as a young child can answer "amen" they have a portion in the World that is Coming. And the gates of Gan Eden are opened for one who answers "amen" with all her power (Shabbat 119b).

13th Century -circle of R Moses deLeon

Soncino translation

...for whoever does not know how to honour the name of his Master were better not to have been born. R. Judah says that these words apply to one who does not answer “Amen” with devotion, since we have learnt: Greater is he that answers “Amen” than he that says the blessing. [T.B. Berachoth, 57b.] For so we have explained in the presence of R. Simeon, that Amen draws blessings from the Source to the King and from the King to the Matrona; in the inscribed letters of R. Eleazar, from A to M and from M to N. and when the blessings come to N. from there (continued...)

Soncino translation

they issue forth to higher and lower and spread through all, and a voice proclaims, “Drink from the stream of blessings that so-and-so the servant of the Holy King has sent forth”.

Soncino translation

R. Jose cited the verse: “For my people have committed two evils: they have forsaken me, the fountain of living waters, and have hewed them out cisterns, broken cisterns, that can hold no water” (Jer. II, 13). ‘ “They have forsaken me”, by refusing to sanctify the name of the Holy One, blessed be He, with Amen; and their punishment is “to hew out broken cisterns”, by being taken down to Gehinnom storey after storey till they reach Abadon, which is called “nether”.

But if one sanctifies the name of the Holy One, blessed be He, by answering Amen with all his heart, he ascends grade after grade till he is regaled with that World-to-come which perennially issues forth...

(6) ...May your great name be made holier and greater, in this world that he is destined to renew and to give life to the dead and raise them to eternal life, to build the city of Yerushalayim, and complete the Beis HaMikdosh in its midst. And to uproot idol worship from the Land, return the service of the Heaven to its place, and the Holy One Blessed Be He shall rule in his majesty and splendor in our lives, and in the lifetime of the entire household of Israel, swiftly and in the near future; and say, Amen. May his great name be blessed, forever and ever. Blessed, praised, glorified, exalted, extolled, honored elevated and lauded be the Name of the holy one, Blessed is he- above and beyond any blessings and hymns, Praises and consolations which are uttered in the world; and say Amen. May there be abundant peace from Heaven, and life, upon us and upon all Israel; and say, Amen.