After the capture of Jerusalem in 586 BCE by King Nebuchadnezzar of Babylonia, and the destruction of the Bais Hemikdash, the king allowed an impoverished remnant of Jews to remain in the land of Yehudah and while enjoying a new respite after their earlier oppression they tended the vineyards he gave them. He appointed Gedaliah ben Achikam, of a prestigious family, to be governor over them.
The King of Ammon, however, was hostile and envious of this situation, and sent Yishmael ben Netaniah to assasinate Gedaliah. Gedaliah did not pay any heed to the rumors he heard of Yishmael's murderous intent (as he refused to listen to lashon hora) and he was murdered together with many of the Jewish citizens of Yehudah while the remaining Jews fled to Egypt.
As a result of Gedaliah's death the final vestiges of Judean autonomy after the Babylonian conquest were destroyed; the remaining Jews were dispersed; the land remained desolate.
Whether Gedaliah was slain on the third of Tishrei or on the first, as some historians claim, we fast on the third day of Tishrei. Do we fast just to burn off the excess calories ingested on Rosh HaShana? Of course not, but rather the aim of this taanis is to establish that the death of a tzaddik is equated with the destruction of the Holy Temple and just as a fast day was ordained upon the churban Bais HaMikdash so, too, a fast was established on the death of the tzaddik, Gedaliah.