A fast can only be effective if Jews who have sinned [and are now repenting] participate, as symbolized by the chelbanah in the ketoret [Temple incense]: Keritot 6b
The Reward for Fast Days is in giving Tzedakah to support Paupers afterward: Berachot 6b (see Rashi)
Whether one is permitted to cause pain to himself, by voluntarily refraining from something: Nedarim 10a
Fasting as a form of prayer: Sotah 12a
The value of committing to fast at a future date, even if one cannot fast right now: Taanit 8b