The prophets established a fast day on the tenth of Tevet, because that is when Nevuchadnetzar King of Babylonia came with his army to besiege Jerusalem, and that marked the beginning of the troubles which ended in the destruction of the First Temple and the exile of the Shechinah. Granted, during the Second Commonwealth, the siege began on a different day; nevertheless, the initial destruction of the Beit HaMikdash and abolishment of Jewish kingship occurred on the tenth of Tevet.
Once the tenth of Tevet was already established as a fast day, the rabbis added two other sorrowful events, which occurred around that date, to the character of the day: the death of Ezra the Scribe, on the ninth of Tevet, and the translation of the Torah to Greek, on the eighth of the month. The Chief Rabbinate of Israel also expanded upon the significance of the day, establishing it as a day of general mourning (Yom HaKaddish HaKlalli) for the holy souls who were killed in the Holocaust, whose date of death is unknown.
Our Sages say that Ezra the Scribe was worthy to have the Torah given through him, had Moshe Rabbeinu not preceded him (Sanhedrin 21b), implying that he was second only to Moshe. Ezra the Scribe enacted ten fundamental decrees (Bava Kama 82a), and by doing so, he inaugurated the tradition of the Sages of the Oral Law. Chazal further state that Ezra the Scribe, who ascended from Babylonia to build the Second Temple, is actually the prophet Malachi (Megillah 15a). That is to say: on the one hand, he is the last of the prophets, who are associated with the Written Law; while on the other hand, he is the first of the Sages of the Oral Law. Thus, he is a great Torah scholar who serves as a transitional link between the Written and Oral Torahs. And just like Moshe, he cared for Klal Yisrael and bore the burden of leading them. He was among the leaders of the aliyah from Babylonia and one of the builders of the Second Temple.
Afterwards, when the Greeks came to power, they issued a harsh decree against the Jews, forcing them to translate the Torah into Greek. That day was as calamitous for the Jews as the day upon which the Golden Calf was made, for the Torah belongs to the Jewish people and translating it into Greek blurred its uniqueness, giving the impression that anyone can engage in it. This occurred on the eighth of Tevet, causing darkness to descend upon the world for three days. Therefore, on the fast of Asarah B’Tevet we mention this painful event, as well.
Our master and teacher, R. Tzvi Yehudah Kook zt”l, said that we must rectify these three issues on the tenth of Tevet: 1) In response to the siege of Jerusalem, we must strengthen the walls of the city and build up the Land both spiritually and physically. 2) In response to the death of Ezra, we must enhance and glorify the Torah, while engaging in the ingathering of the exiles, as Ezra the Scribe did. 3) In response to the translation of the Torah into Greek, we must restore our genuine Jewish spirit and culture and uproot all the evil “spirits” that entered our culture throughout the exile, when the nations ruled over us.