The origin of this custom is unknown. It stands in contrast to the traditions surrounding other Jewish holidays, which often call for eating meat. Since it was often more expensive than other foods, meat was reserved for celebratory events.
Rabbinic tradition has suggested a number of reasons for this custom. Chapter 4, verse 11 of The Songs of Songs compares Torah to honey and milk. As a result of this lovely comparison, it is typical on Shavuot to eat blintzes, cheesecake, and noodle kugels.