This is the first mention that we see of the tenth day of Tevet as a fast day. As the tenth month counting from Nissan (which is referred to as the first month) is Tevet, the fast of the "tenth month" is the Fast of the Tenth of Tevet.
On the tenth day of Tevet, the wicked Nevuchadnezzar, the king of Bavel, laid siege to the city of Jerusalem.
"And it came to pass in the ninth year of his reign, in the tenth
month, in the tenth day of the month, that Nevuchadnezzar, king of Bavel came,
with all his troops, upon Jerusalem...."
Melachim (Kings) II 25:1
At the time of this siege, there were already Jews who were in exile in Bavel, where the prophet Yechezkel (Ezekiel) lived. G-d came to Yechezkel in a prophecy, to inform him of the siege of Jerusalem.
"And in the ninth year, in the tenth month, on the tenth day of
the month, the word of G-d came to me, saying 'Son of man, write the name of the day,
of this same day - the king of Bavel has laid siege on Jerusalem on this very
Yechezkel (Ezekiel) 24:1
At that time in Bavel, there were false prophets who spread the word that the Temple in Jerusalem would never be destroyed and that the Jewish nation would never be exiled from the land of Israel. While these "prophets" spread their message of falsehood, the true prophets of G-d, namely Yirmiya (Jeremiah) exhorted the nation of Israel to repent and turn from their evil ways. He warned that the consequences of the nation remaining sinful were dire - the Temple would indeed be destroyed and the Jewish nation would indeed be exiled from the land of Israel. However, the nation did not heed the warning of Yirmiya, and they remained unrepentant and steeped in their sinful ways.
When the siege of Jerusalem finally occurred, as the true prophets of G-d had warned, G-d wanted the people to immediately get the message that the prophet Yirmiya was truthful and correct, and that all of his prophecies were the truth. Therefore, G-d came to Yechezkel on the day that the siege took place and related to him the day's events. All of the people in Bavel would immediately hear from Yechezkel about the tragic event, the event that the "prophets" they listened to said would never happen. The people of Bavel would see that they erred by not repenting and by ignoring the word of G-d. Yechezkel was therefore told by G-d to make note of this date, so all would recognize G-d's true messengers, know of the events and the of the forthcoming consequences which they thought would never happen.
We fast on this day because it marks the beginning of our sorrows, the first event in a chain which resulted in the eventual destruction of Jerusalem and the Temple, and the exile of the nation of Israel.