Numerous sources emphasize the great importance and value of answering to Kaddish.
In Masechet Berachot (6b), the Gemara teaches that when Hashem comes into the synagogue and sees that there are fewer than ten men present, "Miyad Hu Ko'es" – He immediately becomes angry. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) raised the question of why the Gemara adds the word "Miyad" – "immediately." What is added by telling us that G-d's anger is aroused instantly? The Ben Ish Hai answers by suggesting that "Miyad Hu Ko'es" means that Hashem grows angry because of "Yad" – the letters Yod and Dalet. The letter Yod equals 10, alluding to the minimum of ten Kaddishim which one should hear and respond to each day, and Dalet equals 4 – referring to the four recitations of Nakdishach which a person should hear and respond to each day. When people do not come to the Minyan, Hashem becomes angry – even though the people can pray privately, because they cannot respond to Kaddish or Nakdishach.
The Gemara in Masechet Sota (49a) states that although the world's condition has been worsening progressively since the Bet Ha'mikdash was destroyed, it is sustained in the merit of the "Yeheh Shemeh Rabba" response to Kaddish, and of the recitation of the Kidusha De'sidra (a section of the U'ba Le'sion prayer). Moreover, the Gemara in Masechet Shabbat (119b, according to one version of the text) teaches that if a person was deemed worthy of seventy years of suffering, he can have the decree rescinded in the merit of responding "Yeheh Shemeh Rabba" with all his strength. The Gemara further states that the merit of this response can bring a person atonement even for the sin of idolatry. Another passage there in the Gemara teaches that if a person responds "Yeheh Shemeh Rabba" with all his strength, the gates of Gan Eden are opened for him. Similarly, the Sefer Hasidim (Rabbenu Yehuda Ha'hasid, Germany, 1150-1217) writes that one who regularly responds "Amen" in this world earns the privilege of doing so also in the world to come. This is alluded to in the verse in Tehillim (89:53), "Baruch Hashem Le'olam Amen Ve'amen" ("Blessed is G-d forever, Amen and Amen"). The phrase "Amen Ve'amen" alludes to the response of "Amen" both in this world and the next.
Another important source is the Gemara's teaching in Masechet Berachot (3a) that when Jews gather in the synagogue and declare, "Yeheh Shemeh Rabba," Hashem exclaims, "Fortunate is the king whose subjects praise him this way!"
The Bet Yosef cites the Zohar as explaining that Kaddish is recited in Aramaic, instead of Hebrew, because it has the unique power to oppose the Kelipot ("shells," the harmful spiritual forces). We use the inferior language, Aramaic, so we can attack the Kelipot in their language, as it were, and this has the effect of eliminating the forces of evil from the world.
Tosafot (Shabbat 119b) cite a story from the Midrash about Rabbi Yishmael Kohen Gadol, who was shown how the dreadful punishments that are decreed upon Beneh Yisrael are avoided in the merit of the response of "Yeheh Shemeh Rabba." And the Zohar states that the sign of a great Torah scholar is if he fervently looks for opportunities to respond to Kaddish. If a person rushes out of the synagogue before the final Kaddish, then even if he is a scholar, he cannot be considered a true Talmid Hacham.
The Gaon of Vilna (1720-1797) writes that those who answer "Yeheh Shemeh Rabba" will be spared the suffering from the upheavals that will occur before the arrival of Mashiah. The Mishna Berura cites a passage from the Midrash describing Hashem's reaction when Jews assemble to learn Torah and then recite Kaddish – He turns to angels and exclaims, "See how My children praise me!"
Importantly, however, Rav Moshe Zakuta (1625-1697) writes that one must respond "Amen" with Kavana (concentration). If a person answers mindlessly, without paying attention, then he is included, Heaven forbid, in G-d's warning, "U'bozai Yekalu" – that those who disgrace Him will be shamed (Shemuel I 2:30).
It is told that Rav Mordechai Gifter (1915-2001), the esteemed Rosh Yeshiva of Telz in Cleveland, once traveled with eight students to Toronto for a wedding. They were altogether nine men, and thus could not form a Minyan, but they assumed that they would have time upon arriving in Toronto to join a Minyan for Minha. As it happened, however, the plane made an emergency landing in some small town between Cleveland and Toronto. The group needed to recite Minha there, despite not having a Minyan. To their astonishment, a worker in the airport approached them as they were starting to pray and informed them that he was Jewish and wished to join them. He could not even read Hebrew, but he told the group that he wanted to recite Kaddish, and he needed their help. They made a Minyan, and helped him recite Kaddish.
Afterward, Rav Gifter spoke to him and asked why he, a Jew without any religious background, wished to pray and recite Kaddish.
The man explained that his father passed away several days earlier. The night before he met this group in the airport, his father came to him in a dream and told him he needed him to recite Kaddish for him. The man asked his father how he could recite Kaddish, as he lived in a town without a Jewish community.
"Don't' worry," his father said, "tomorrow I'll send you a Minyan so you can recite Kaddish."
This story demonstrates how everything happens for a purpose, and that nothing is random – but additionally, it teaches us the importance of Kaddish, and the great benefit it brings to the soul of the deceased when the children recite Kaddish.