The sefer Chayei Adam states that the Anshei Kineses HaGedola (Men of the Great Assembly) decreed that Kaddish be said daily. The source is a pasuk in Parshas Emor, 'I will be sanctified amongst the Children of Israel' (Vayikra 22:32).
The Tur (Orach Chaim 56) cites a pasuk in sefer Yechezkel describing milchemes Gog and Magog. In the midst of that final war, Hashem will reveal Himself, 'I will be exalted (vi'his'gadilti) and I will be sanctified (vi'his'kadishti) and I will make Myself known in the eyes of many nations; and they will know that I am Hashem' (Yechezkel 38:23). After the initial revelation His Name will be exalted, as the pasuk states, 'On that day Hashem will be One and His Name will be One' (Zechariah 14:9). His Name can only be complete after Amalek is defeated in that final war as the pasuk tells us, 'For the hand is on Hashem's throne.' Shemos 17:16 - see Rashi there who says For there is a hand on the throne of the Eternal: The hand of the Holy One, blessed be He, was raised to swear by His throne, to have a war and [bear] hatred against Amalek for eternity. He, swore that His Name will not be complete and His throne will not be complete until the name of Amalek is completely obliterated. And when his name is obliterated, the Divine Name will be complete, and the throne will be complete. Therefore by saying Kaddish and answering Amen Yehei Shemay Rabbah we are declaring that great day will come, when Hashem will reveal His greatness to the entire world and destroy Amalek.
Rav Yom Tov Lipman Heller writes in Ma’adanei Yom Tov (citing the text of the Gemara Berachos Daf 3 which reads: Rabbi Yehoshua ben Levi said that whoever answers [Kaddish], Amen, Yehei Shemay Rabba Mevorach, responding with all his strength [up in the Heavens], they tear up the judgments of seventy years against him. That is why the Birkas HaTorah was established using the Divine four letter name of loving kindness, YH”V”H.)
The definition of Shemay Rabbah Mevorach is that while we are praying, we are asking that the Divine Name of Shem Yud-Hay should become whole through our redemption as we say that Hashem’s throne and Name are incomplete.
Also [a second request is that we are asking, May] His Divine Name be blessed – mevorach. The Tosafos in the commentary to Daf 3 explain this answer differently, interpreting Yehei Shemay Rabbah Mevorach as [one single prayer], ‘May His great Name be blessed.’
Rashi explains the meaning of “a decree of seventy years” as a decree that may have been made in his youth. I believe the meaning refers to a person’s entire lifetime, as the pasuk in Tehillim (90:10) tells us that an average lifetime is seventy years. The Gemara is telling us that all the decrees and harsh judgments that have been decreed or leveled upon a person throughout his entire life shall be canceled, nullified, ripped and torn up.
In his commentary, Divrei Chamudos, Rav Heller continues:
I cited two explanations for Amen, Yehei Shemay Rabba in my commentary Ma’adanei Yom Tov and there is a practical difference in outcome between the two different interpretations. According to the first explanation, that Amen, Yehei Shemay Rabbah is a prayer that the name Yud-Hay be increased and completed, the glosses of Hagahos Ashri cites the Ohr Zarua that we should not interrupt or pause in between the words Shemay and Rabbah since it is one fluent idea. However, you may pause between the words Rabba and Mevorach since, according to this interpretation, the word mevorach is a separate prayer. It’s a new request that, besides completing the name Ya”h into the four letter name YH”V”H, we are also asking that it be blessed forever for olam haba – mevorach le’alam u’l’olmei olmaya.
According to the second interpretation of the Tosafos that explain that Shemay Rabba means His great Name, and there should be no pause between Rabba and Mevorach either since it is all one single cohesive prayer. It does not mean [as the first opinion] that His name should be great and completed, rather that His great Name be blessed. . .Furthermore, the Beis Yosef cites a midrash and concludes that we should not pause or interrupt between Olmaya and Yisbarach (Ma’adanei Yom Tov and Divrei Chamudos on Rabbeinu Asher – The Rosh Berachos 3:8-9 and 3:61).
