On the third of Tishrei, Gedalyah ben (son of) Achikam was killed. After the First Temple was destroyed and most of the Jews were exiled to Babylonia, the king of Babylonia appointed Gedalyah ben Achikam as the leader of the remaining Jews of Judea. [Am Yisrael as a whole] pinned its hopes on these remnants, [reasoning] that if they managed to establish themselves in the Land, they could rebuild the Beit HaMikdash and reestablish Jewish sovereignty, together with those who [eventually] return from Babylonia after the seventy years [of exile]. Indeed, for some time it seemed that the meager population that remained in the Land was recovering from the churban and cultivating its fields and vineyards.
However, the king of Ammon, who wanted to eradicate the remnant of Israel, sent Yishmael ben Netanyah to murder Gedalyah. Yishmael had an additional, personal reason to carry out the scheme: he was a scion of the Judaic dynasty, and he thought that he deserved the privilege of ruling over Judea, instead of Gedalyah ben Achikam. Several of Gedalyah’s generals warned him about Yishmael and even recommended killing him before he succeeded in carrying out his plan. But Gedalyah didn’t believe them, accusing them of maligning Yishmael. It came to pass in the seventh month, that Yishmael son of Netanyah son of Elishama, of the royal seed, along with the king’s captains and ten men, came to Gedalyah son of Achikam at Mitzpah; and they ate bread together there at Mitzpah. Then, Yishmael son of Netanyah and the ten men who were with him arose and struck Gedalyah son of Achikam son of Shafan by the sword, and killed him… along with all the Jews who were with him… and the Chaldeans who were found there(Yirmiyah 41:1-3). This extinguished the last ember of Jewish rule in Eretz Yisrael and consummated the exile. This is why the prophets declared [the third of Tishrei] a fast day.
Actually, there is some doubt as to when exactly Gedalyah was killed. The Talmud states that the murder happened on the third of Tishrei (Rosh HaShanah 18b; Yerushalmi, Ta’anit 4:5). However, some Rishonim explain that he was killed on the first of Tishrei, but since the holiday of Rosh HaShanah comes out on that day, [the Rabbis] postponed the fast day until the third of Tishrei.
The Sages also state that the institution of Tzom Gedalyah demonstrates that the death of a righteous individual is tantamount to the destruction of the House of our God (Rosh HaShanah 18b).
. This is what Rabbeinu Yerucham writes (Netiv 18, beginning of section 2). In accordance with this, the Ibn Ezra and Radak explain that the word chodesh (month) [see the verses above] implies the first of the month, because that is when the month is renewed. [The Rabbis] simply postponed the fast until the third of the month. Even though [the Jews] observed only one day of Rosh HaShanah at that time, [the Rabbis] did not want to [require us to] fast immediately following a holiday. Therefore, they pushed the fast off until the third of Tishrei. And even after [the Jews] began keeping two days of Rosh HaShanah, the fast remained on the third of Tishrei. See further sources on this in Torat HaMo’adim 1:2 (p. 8). The author of Responsa Rosh Yosef holds that since Tzom Gedalyah is observed on a day other than its true date, its laws are more lenient [than those of the other fasts]. Hence, one who celebrates a brit milah [on the third of Tishrei] may eat a festive meal and need not make up the fast. The Taz (sec. 549), however, dismisses his words, as does the Bi’ur Halachah (ibid.), based on the Ritva. Their reasoning: even if we say that Gedalyah was actually killed on the first of Tishrei, since the fast was established on the third of the month, it is not considered “postponed.” Rather, that is its [proper] date.