To our limited vision, it seems that the days commemorated by fasts are sad, lonely days. But the Rebbe explains that when Mashiach
comes things will be revealed in their true reality. And then we will see that these days were not as bad as they seemed. On the contrary, we will see how these were days of tremendous potential, of closeness and holiness. The same applies to the Tenth of Teves.
The number ten indicates a certain level of sanctity, of wholeness and perfection in holy matters. There are Ten Sefirot, (Divine Emanations), and Ten Utterances through which the world was created. There are Ten Commandments, and ten levels of holiness in the Land of Israel. It is no coincidence that Yom Kippur, the holiest day of the year, also falls on the 10th of the month, the 10th of Tishrei. "OK," you say, "Tishrei is Tishrei, but Teves? What significance does Teves have?" As you know, we count the months from Nissan, which the Torah calls the first month. What month is Teves if Nissan is the first? The tenth! So the tenth of Teves is the 10th of the 10th. An indication of an extra level of kedushah that is not found even in the festivals!
In the Torah, the period of galus (Exile) and especially the end of galus is often compared to the situation of a woman about to give birth. As the actual process of birth comes closer, the pains are more frequent and more difficult to bear. However, soon afterwards, the pain is just a past, vague memory, and the day the baby was born is now remembered as a joyful day. As the years go by, the pain becomes less and less prominent, and the joy and nachas will hopefully outweigh the memory of the pain and difficulty of giving birth. And that is the idea of galus. We are now experiencing the birthpains, the contractions. It is difficult, but they are temporary, only a small price that we must pay for the joy and the great blessing that are going to come out of them. Again, if I had the time I would go into it, but this is one way of looking at a fast day in a positive way, and using it studying and trying to be better and more Jewish, knowing that when it will be a Yom-Tov and Mashiach will be here, it will all be forgotten.