There are many rabbinically posited explanations for the custom of eating dairy foods on the morning of Shavuot, some more fanciful than others. But the real reason may have much more to do with practicality and non-Judaism-related local custom than to any holy reason rabbis might imagine.
In his Moadim Uzmanim (1958), Rabbi Moshe Sternbuch, the #2 rabbi in the über-haredi Edah Haredi umbrella organization, tries to explain the mysterious custom of eating dairy foods on Shavuot.
As Ha'aretz cites him, Sternbuch claims that before the giving of the Torah, the ancient Hebrews could not drink milk or use it in any food because it violated the 6th of the 7 Mitzvot of the Children of Noah:
…Citing the Talmud (Bechorot 6b), Rabbi Sternbuch says that before the Torah was given, the Israelites were not allowed to eat dairy products, since these were considered “a part of a live animal”. But once the Torah was handed down, the passages containing the phrase “Land of Milk and Honey” (e.g., Exodus 3:8) made their consumption permissible.
Thus we eat dairy products to commemorate the fact that on Shavuot, God allowed us to eat these.…
But rabbinic Judaism believes that the ancients, not just the ancient Hebrews, all knew of those seven mitzvot and observed some or even all of them.
So why, then, don't we find other cultures who shunned milk for that reason?
Shouldn't the Indians or the Turks or the Egyptians or the Bedouins or someone in the region also had this tradition?
At any rate, the custom to eat dairy on Shavuot is first mentioned in Rabbi Isaac Tyrnau’s Sefer HaMinhagim (“Book of Customs”) which was written in the late 1300s in Austria or Romania. [Update – –The first mention appears to be a couple hundred years earlier, but still in Europe and in areas where cattle were plentiful and relatively cheap.]
And that may provide the answer to the riddle.
Unlike the arid Middle East, in Europe dairy cattle were very common because there was lots of good land to pasture them on and cow's milk consumption was much higher there than in the Levant.
Jews who came to Central and Eastern Europe had easy access to cheap cow's milk – a good protein source – for the first time and likely began consuming lots of it (except, of course, for those who were lactose intolerant).
On Shavuot, there was (and still remains in many communities) a custom to pray very early on the morning of Shavuot, preferably with the sunrise, and follow it with the morning holiday meal.
For Jews who regularly ate dairy products, a meal at 6 am or 7 am would have rarely contained any meat. But it would very likely have contained fresh milk from the morning's milking – which was customarily done by non-Jews on Jewish holidays and on the Sabbath.
Because Jews were in the synagogue praying, on Shavuot morning they could not watch that milking and likely would not have had access to fresh milk, because no Jew would have been available to watch the milking process.
But they would have had access to other cow's milk-derived foods like cheese, blintzes, butter and the like.
On a regular Sabbath or holiday, for most Jews, their morning meal would have taken place sometime just before noon (anywhere from perhaps 10 am onward).
But on Shavuot they would have finished praying much earlier. And because many stayed up the entire night before studying Torah in honor of the giving of the Torah, which rabbinically took place on Shavuot, they would have tired, as well. And if they fell asleep, they could easily sleep through the proper time of the morning meal – remember, this was before alarm clocks or even clocks.
So the practical thing to do was pray, immediately make kiddush, eat a light meal and then, if sleep overcame them, it overcame them.
But the key idea here is "light meal."
Eating a heavy meat meal would easily put to sleep many people who had stayed up all night and then exerted themselves in prayer at dawn.
But a much lighter meal whose protein source was dairy? That would be much less likely to do so.
And that, I think, may be the actual origin of the holy custom of eating dairy foods – primarily, you'll note, of eating cooked or baked dairy foods that could be prepared a day in advance and kep without spoiling at a time when no refrigeration existed – on Shavuot morning.