(There are many online sites that give reasons for the Jewish custom of eating dairy on Sh’vuot. As can be seen, there’s no single explanation. Rather, multiple reasons are suggested, none of which are exclusive. It might actually be of greatest help to our kavannah to review in our minds — and in our conversation — all of the various reasons we’re given.)
“On Shavuot we celebrate the giving of the Torah. This gift was one of complete compassion and loving-kindness, for with the Torah we were given the tools (i.e. knowledge, commandments) to connect with the infinite and otherwise unknowable Creator.
[Dairy foods such as cheesecake, cheese blintzes, and cheese kreplach among Ashkenazi Jews; cheese sambusak, kelsonnes (cheese ravioli; i.e. like the Italian ‘calzone’), and atayef (a cheese-filled pancake) among Syrian Jews; kahee (a dough that is buttered and sugared) among Iraqi Jews; and a seven-layer cake called siete cielos (seven heavens) among Tunisian and Moroccan Jews are traditionally consumed on the Shavuot holiday. Yemenite Jews do not eat dairy foods on Shavuot.*]
Dairy foods are associated with the loving, nurturing generosity exemplified by a mother nursing her baby. It is this supreme love that we connect to on the anniversary of the giving of the Torah on Mount Sinai. New beginnings and connecting to the Source is what Shavuot is all about.
Here are a number of other reasons for the custom to eat dairy on the first day of Shavuot:
1. Chalav—the Hebrew word for milk—has the numerical value (Gematriah) of 40 reminding us the number of days and nights that Moses remained on Mt. Sinai.
2. One of the eight different names for Mt. Sinai is “Gavnunim,” which means white like cheese.
3. The words in the Torah referring to the Shavuot holiday offering are “Minchah chadashah l’Hashem b’shavuotaychem,” which are also an acronym for the Hebrew word m’chalav—”from milk.”
4. When the Jews received the Torah on Shavuot they were commanded only to eat meat which was ritually slaughtered. Since none of their meat was previously slaughtered [according to the laws of Kashrut, which had just been given] and the Torah was given on Shabbat — when it is forbidden to slaughter animals — they were forced to eat dairy for the rest of the day.
5. Shavuot is the completion of a spiritual process that we begin on Passover, and their respective holiday offerings represent the stages of this process. At the Passover Seder we have two cooked dishes to commemorate the two offering brought on Passover in the Temple times. To connect the two holidays, we eat two cooked foods on Shavuot as well—one meat and one dairy.
6. Two loaves of bread were offered in the Holy Temple on the holiday of Shavuot. To commemorate this offering we eat two meals on Shavuot; one dairy and one meat (eating meat is mandatory on every festival).” 
(And yet, I think that once a minhag or custom evolves, it becomes an essential part of the overall mitzvah itself. For whatever reason eating dairy is singled out for practice on Sh’vuot, the doing of it in itself helps make that particular holiday unique. Just as with eating matzah on Pesach or eating something deep-fried on Chanukah, it can become a devotional act in itself, if our kavannah is on the holiday and not just on enjoying the food. Observing, we join ourselves — our finite selves — to the larger Self expressed through tradition. We need not look for a literal meaning, as much as for how eating dairy makes Sh’vuot something “special” and links us to something larger than ourselves, which is also the higher part of ourselves.)