The Kaddish has only praised God so far. So which prayers are we asking Him to accept?
Old sources noticed this too. Until this line, the Kaddish has only praised God. It has not asked for anything. The Kol Bo (a medieval collection of Jewish law and custom) points at the puzzle. The Kaddish, he writes, holds no prayer. If we said "may their prayer and request be heard," the angels would be surprised and ask what prayer these people want heard, since they said only praise. So, he says, we say the line in Aramaic, so they do not notice.
One answer is that the line looks back at the Amidah, the standing prayer you just finished. Terumat HaDeshen (Rabbi Yisrael Isserlein, a rabbi in Austria in the 1400s) writes that this Kaddish goes back to the Amidah "since it says titkabal," and that after verses and words of Torah nobody says it. The Rema (Rabbi Moshe Isserles, who added Ashkenazi practice to the Shulchan Aruch, the main code of Jewish law) uses this in law: this Kaddish "belongs to the prayer." The Mishnah Berurah (a central commentary on daily Jewish law, written around 1900) says the same.
Another answer is that the Kaddish is itself a prayer. Likkutei HaPardes (a collection attributed to Rashi's school) asks "what prayer did they make?" and answers: the prayer that God's name be made great, a prayer for redemption. The Tur (a law code from the 1300s) records that some read the opening that way.
Tosafot (the medieval French and German commentators on the Gemara) argue against it. In their reading, "May His great name be blessed" is one sentence. It does not ask for God's name to become complete. The Tur agrees with them.
Rabbi Yehuda Henkin, in Benei Banim, his modern book of answers on Jewish law, lines up the two sides. If the opening is only praise, "their prayers" must mean the Amidah. If the opening is a prayer, it can mean the Kaddish itself. The Rema and the Mishnah Berurah treat the line as pointing back to the Amidah. The sources do not say which one was meant when the line was first added.
Kol Bo 7:17,Terumat HaDeshen, Part I 15,Shulchan Aruch, Orach Chaim 55:3, Rema,Mishnah Berurah 55:5,Likkutei HaPardes, Interpretation of the Kaddish,Tur, Orach Chaim 56,Tosafot on Berakhot 3a,Benei Banim, Volume II 4:18I already asked God to accept my words in my own Amidah. Why does the leader ask again?
Near the end of your own Amidah you say a verse from Tehillim: "May the words of my mouth and the thoughts of my heart be accepted before You." That is one person asking, in the singular. The Elokai Netzor unit tells its story. Titkabal is a different request. It speaks of the prayers "of all the House of Israel." Yours is in there with everyone else's.
A stranger detail is about the leader himself. He repeats the Amidah out loud. The Shulchan Aruch says that when he does, he says the opening line, "Hashem, open my lips," but he does not say the closing verse, "May the words of my mouth be accepted." The Magen Avraham (a commentary on the Shulchan Aruch from 1600s Poland) gives the reason: the leader relies on "titkabal tzlotehon." For him, this line of the Kaddish does the job of that verse.
Not everyone agreed. The Magen Avraham adds that the Shelah (a book by Rabbi Yeshayahu Horowitz, written in the early 1600s) says the leader should say the verse after all.
This also explains when the leader is finished. Peninei Halakhah says he does not step back at the end of the repetition. His prayer is not completely over until this Kaddish, where he asks that our prayers and requests be accepted. He takes his three steps back at the end of it.
Tehillim 19:15,Shulchan Aruch, Orach Chaim 123:6,Magen Avraham 123:14,Peninei Halakhah, Prayer 17:14Why does a prayer about being heard end with a wish for peace?
The last two lines are the extra pieces that make a Kaddish "full." Peninei Halakhah says they add a request for peace and good life for us and for all Israel, and the Kaddish closes with "He who makes peace."
The Arukh HaShulchan gives a reason that fits this spot. He says the name "half Kaddish" misleads some people. The Kaddish itself is complete at "that are said in the world." At the end of the prayers we add "titkabal" and the wish for peace. His reason: peace is the pillar of the world, and now that the prayer is over and we have to go out into the world, we ask God for peace among us.
He also separates the two peace lines. First we ask that there be peace among us. Then we ask that, if a quarrel breaks out anyway, God help us make peace the way He does in the heights between opposites. The Kaddish DeRabbanan unit tells what the Tur says those opposites are.
The Sim Shalom blessing at the end of the Amidah tells the Mishnah's teaching that peace is the only vessel that can hold blessing.
Likkutei HaPardes (a collection attributed to Rashi's school) notices something else. The Amidah ends with peace, and so does Birkat Hamazon (Grace after Meals). So the Kaddish ends the way big prayers end.
Peninei Halakhah, Prayer 23:7,Arukh HaShulchan, Orach Chaim 56:8,Likkutei HaPardes, Interpretation of the Kaddish