The Kaddish prayer is supposedly inspired by Ezekiel 38:23 which ends in a hope that one day all nations will sanctify the name of G-d. These words were probably especially meaningful to Ezekiel, who was exiled from his home (with 3,000 other Jews) for his religious beliefs. He later went on to prophesy the destruction of Jerusalem. The end of his life is something of a mystery and is not described in any detail in the Torah or other religious texts.
Today the Kaddish continues to inspire Jews around the globe. It is a unifying prayer that praises G-d and connects those reciting it to the long Jewish tradition. Every time the Kaddish is said it echoes through time and becomes part of the Jewish tradition. This is why the Kaddish is said so often; after study, after prayer, and during the mourning period.
The Kaddish is a prayer of connection. It connects Jews to other Jews and to G-d. That is why it can bring so much comfort to the grieving. It connects them to their deceased loved one. It lets them know that their soul is with the soul of other Jews. It also lets them connect with the fact that Jews throughout time have been mourning, and growing stronger from it.