At Mount Sinai, G-d gave the Jewish people the Ten Commandments as well as the rest of the Torah. Rabbi Saadya Gaon, in one of his works, explains how the 10 Commandments contain all 613 commandments of the Torah. Thus, on Shavuot, they also received the laws of Kashrut. As a result, all their utensils from before were non-Kosher. Without the use of their utensils they had no choice but to eat dairy on the day of Shavuot. To commemorate this, we eat dairy on this day.
When Jews received the Torah, they were considered as newborns. Just as an infant is nourished through milk, we too eat dairy on Shavuot to symbolize the birth of the Jewish nation.
Shavuot occurs exactly three months after the 7th of Adar - Moshe's birthday. The Torah tells us that when Moshe was three month old, his mother placed him in a basket in the river. Pharaoh's daughter found him, pulled him from the river and gave him to an Egyptian woman to nurse. Moshe refused to nurse from the Egyptian and as a result, he was given to his mother, Yocheved, to nurse. On Shavuot, we eat dairy foods to recall this episode in Moshe's life, when he was returned to his mother.
Milk in Hebrew is Chalav. The numerical value of Chalav in Hebrew is 40 (8+30+2=40). This alludes to the 40 days Moshe stayed on the mountain receiving the Torah.
The 613 Mitzvot are comprised of 248 positive Mitzvot and 365 negative Mitzvot (prohibitions). The 365 prohibitions represent the 365 days of the year. The Mitzvah of not mixing meat and milk corresponds to the day of Shavuot. To show that we observe this mitzvah, we begin the Shavuot meal with dairy, wait the appropriate time and then eat meat.
When Moshe went up to receive the Torah, the angels said to G-d, "Why give the Torah to man? Better keep it in heaven for us!" G-d told Moshe to reply to them. Moshe said to the angels, "When the three angels came to Abraham disguised as people Abraham gave them milk and meat, which they ate. How can G-d give you the Torah which prohibits eating meat and milk together? You had the Torah in Heaven, yet you weren't careful with this mitzvah at Abraham's house!" Because this mitzvah was a factor in getting the Torah on Shavuot, we show that we are careful not to mix meat and milk together. Instead, we eat first the dairy and only after, eat meat!