Naso: 5783 - Rabbi Shimon Zonnenfeld explains why dairy foods are eaten on Shavuos. The idea is shared at the conclusion of Rabbi Betzalel Mizrachi's Daf Yomi class in Nachlah. Watch:

00:02 Speaker 1 One of the customs we observe on Shavuos—a custom everyone observes—is eating dairy foods. What is the source of this custom? There are several explanations. One is that when the Torah was given, mixing meat and milk became forbidden. Suddenly, it turned out that they had cooked meat and milk in all their cooking utensils, so they could no longer use them. They ate dairy foods that did not require cooking. That is one explanation. But a question is asked: Hadn’t they been eating manna until then? That was their food, not meat or dairy. The answer is simple: In addition to the manna, they also had cattle and various other things. So they had milk and meat as well. But there is another question: Couldn’t they have koshered their utensils?

01:12 Speaker 2 One day.

01:13 Speaker 1 Yes, no problem. They could kosher the utensils and then use them to cook other things, including meat. But the answer is that the Gemara says in Tractate Shabbos that everyone agrees the Torah was given on Shabbos. Since the Torah was given on Shabbos, they could not kosher the utensils.

01:37 Speaker 2 Ah, wow.

01:39 Speaker 1 And there is a hint to this—

01:40 Speaker 2 [unclear]

01:42 Speaker 1 There is a hint to it as well in the verse, “Honey and milk are under your tongue.” This is also one of the explanations for why we eat dairy foods. The final letters of the first three Hebrew words, “honey and milk are under,” spell Shabbos. Since it was Shabbos, they could not kosher the utensils, so they had to eat foods that did not require cooking. That is where the custom of eating dairy foods specifically on Shavuos comes from. May we merit to receive the Torah with great joy.