What is the fast about? The Prophets and Sages instituted it after the destruction of the First Holy Temple, for on this date the Babylonian King Nevuchadnetzar and his armies began a siege upon the holy city.
After the destruction of the First Temple, the Prophets and the Sages of Israel legislated fasting on the Tenth of Tevet, for it was on that day that Nebuchadnezzar, the King of Babylon, and his legions placed Jerusalem under siege.
After the First Temple was destroyed, the Prophets instituted fasts marking the tragic events surrounding the Destruction and the ensuing exile of the Jewish People.
This step was taken in order to prompt the nation to grieve and mourn over the Destruction and Exile.