Excerpted from "SHAVUOT" - observance, laws, significance. Published by ArtScroll/Mesorah Publications Ltd., Brooklyn, NY. artscroll.com
Many reasons are offered for the custom of eating dairy on Shavuot:
The Ten Commandments encompass all 613 commandments. Thus, when the Jews returned to their tents after the giving of the Torah, they were bound for the first time by the Torah's dietary laws. Therefore, they could not eat meat, for they first had to prepare a proper slaughtering knife, remove the forbidden fats from the slaughtered meat, salt it and procure kosher cooking utensils. All this made it necessary for them to eat dairy, rather than meat, at that time. As a commemoration of this, we eat dairy on Shavuot (Mishnah Berurah 494:12).
Kol Bo writes that it is customary to eat both milk and honey to which the Torah is likened, as it is written, "Honey and milk are under your tongue" (Song of Songs 4:11).
Divrei Yedidyah understands honey and milk as an allusion to the hidden facets of Torah, which are tastier than honey and milk but which must remain "hidden under the tongue," i.e. imparted only to select individuals (see Chagigah 13a).
Why is Torah likened to honey and milk? Honey comes from a bee, which is not kosher, and milk comes from a live animal whose meat is forbidden until the animal is slaughtered. Both honey and milk, therefore, allude to the power of Torah which can transform a sullied soul into one of holiness and purity (from Chag HaShavuot published by Yad L'Achim).
In the writings of Kabbalah, wine and blood symbolize judgment, while water and milk symbolize compassion. Red is also associated with sin and white with atonement, as it is written, "Though your errors will be like scarlet, they will become white as snow; though they will be red as crimson, they will become like wool" (Isaiah 1:18). It is through the purifying process of the Sefirah period that the Jewish people merit Divine compassion on Shavuot, symbolized by the eating of dairy (Magen Avraham 494:6 citing Zohar; see there for further explanation of this concept).
The Talmud (Makkos 23b) states that the 365 negative commandments correspond to the 365 days of the solar calendar. According to Zohar, each, day of the year corresponds to a specific commandment. As mentioned above, in Temple times, the bringing of Bikkurim the first fruits of the seven species with which the Land of Israel:, is blessed, began on Shavuot, and this is one reason why the Torah calls this Festival "Day of the First Fruits." The Torah juxtaposes, "The first of your land's early produce you shall bring to the Temple of the Lord, your God," with, "Do not cook a kid in its mother's milk" (Exodus 34:26), indicating that the day of Shavuot corresponds to the prohibition requiring the separation of meat and milk. To symbolize this, we eat dairy and then meat on Shavuot according to the laws prescribed by halachah (Chidushei HaRim).
As mentioned above, Moses was rescued from the Nile River on the sixth of Sivan, the day on which Shavuot falls. The Talmud relates how Pharaoh's daughter first brought Moses to Egyptian wet nurses, but he would not nurse, for God did not want the mouth that would one day communicate with Him to nurse from a non-Jew (Sotah 12b). Pharaoh's daughter then had Moses given over to Yocheved to be nursed, unaware that this woman was the infant's own mother. The eating of milk foods on the sixth of Sivan commemorates this hidden miracle (Yalkut Yitzchak).
Man cannot live on bread alone, nor can he survive on just water. Yet, one of the miracles of childbirth is that a mother's milk provides her newborn with all the nourishment it needs. In this sense, Torah is like milk, for it encompasses within it all the sustenance that man's soul needs for spiritual vitality and growth. Thus, the milk foods of Shavuot allude to the Torah itself (Imrei Noam).