The author of Piskei Tshuvos writes that Kaddish is “a great and awesome praise established by the Men of The Great Assembly and the Prophets among them.”
It is brought in the Shulchan Aruch (56:1) that one should have intention when asnwering Kaddish. The Mishna Brura explains there that this is because our Sages of Blessed Memory said, “whoever answers 'Amen, Y'hei Shmei Raba Mivorach (May his Great Name Be Blessed)…' with all his strength, they rip up his bad decree in Heaven.”
The Rishonim write that “with all his strength” means with all his intention and all his limbs. I.e., he should say it with his heart and soul, and not just like one who speaks with his lips but his heart is not with him.
Kaddish is a prayer to rectify the desecration of G-d's name and the destruction of the Holy Temple and the scattering of the Jews to the four corners of the world. As the language of Kaddish: May G-d's Name be magnified and sanctified in the world which He created according to His Will. And may He cause His Kingship to rule and may He cause His redemption to sprout and may He bring near (the coming of) His messiah in your lifetimes and in your days and in the lifetimes of the whole Jewish people swiftly and in a time that is soon.
The Tur brings down that Kaddish is based on the words of the prophet Yechezkel (38:23) “[Hashem says] I will be magnified and sanctified in the eyes of the nations.” This verse is talking about the war of Gog and Magog, when G-d's name will be magnified, as it is written, “On that day the Lord will be One and His Name will be One”- Zechariya 14:9.
On a similar note, the Yaavetz writes in his siddur that through saying Kaddish one fulfills the positive commandment, “[G-d says] I shall be sanctified among the children of Israel” – Leviticus 22:32.
When Jews go to shul or to the study hall and answer to kaddish “Amen, may His Great Name be blessed, etc.”, then G-d, so to speak, nods His head and says “Fortunate is the King who is thusly praised in His house (i.e, fortunate all the while that this praise was in the Holy Temple- Rashi). Why did I, their Father, exile My children? Oy to the children who were exiled from their Father's table (i.e, from the Holy Temple where they merited to eat from the offerings- Maharsha)”- Talmud Brochos 3a.
May we merit to fulfill G-d's will and see the redemption soon in our days, Amen.