After the destruction of the Beit Hamikdash, the Rabbis wanted to forbid eating meat. If meat is something that the Rabbis felt could not be eaten due to the fact that we were separated from Hashem, we would think that Shavuot, our “wedding day” with Hashem, would be the holiday of eating fancy meat to celebrate the Torah. However, as we all know, Shavuot is known as “the dairy holiday.” Although some of the meals on Shavuot may consist of meat, we make sure to eat dairy foods as well.
Rav Eliyahu Kitov cites several opinions in his book Sefer HaToda’ah as to why we eat dairy on Shavuot. One is the explanation of Rama (Orach Chaim 494) that Shavuot is a conclusion of Pesach. Just as we commemorate the Korban Chagigah on Pesach by eating two different roasted items, so too on Shavuot we eat two different dishes to commemorate the bringing of the Shtei HaLechem, a dairy meal and a meat meal.
The Sefer Mat’amim points out that by eating dairy on Shavuot, we remember that Moshe was taken from the Nile on 6 Sivan (the day of Shavuot) and that he would only drink the milk of a Jewish woman.
Ta’amei HaMinhagim states that Bnei Yisrael did not eat dairy products before Matan Torah, as they were scared that it would be a violation of Eiver Min HaChai, the prohibition to eat the limb of a living animal, which is one of the Sheva Mitzvot Bnei Noach. Once they got the Torah, however, they realized that they were allowed to eat milk and other dairy products. We therefore we eat dairy products on Shavuot to remember this.
R’ Shimshon of Ostropol points out that the Gematria of the word Chalav, milk, is forty, which is the same number of days that Moshe Rabbeinu spent on Har Sinai. The Geulat Yisrael explains that before Bnei Yisrael got the Torah, they were permitted to eat animals that were not slaughtered according to Halachah, and even meat from non-kosher animals. Once the laws of Kashrut were given to them, however, all of their utensils had to be kashered, so until this was done, they had to eat dairy. We eat dairy foods of Shavuot to commemorate this.