Reason 1: Because there was this whole Shavuot celebration in the Temple
ונוהגין בכל מקום לאכול מאכלי חלב ביום ראשון של שבועות ונראה לי הטעם שהוא כמו השני תבשילין שלוקחין בליל פסח זכר לפסח וזכר לחגיגה כן אוכלים מאכל חלב ואח"כ מאכל בשר וצריכין להביא עמהם ב' לחם על השלחן שהוא במקום המזבח ויש בזה זכרון לשני הלחם שהיו מקריבין ביום הבכורים:
Rema: We have a custom everywhere to eat dairy food on the first day of Shavuot. And it seems to me that the reason is: It is like the two foods that are taken on the evening of Pesach, a reminder of the Korban Pesach and a reminder of the Korban Chagiga, so too we eat a dairy food followed by a meat food, and we bring with them two loaves on the table in place of the Mizbeach, and thus there will be a reminder of the two loaves that were brought on "The day of the Bikkurim."
Reason 2: Say Cheese!
"Jungle Boy," commenter on Haaretz.com
When the Israelites were all gathered around the mountain, Moses arranged for a photographer to take a revolving film of the large population, and when it was time to begin shooting the film, what did he call out? --"Cheese."
Reason 3: No Kosher Dishes!
(יב) {יב} מאכלי חלב - עיין מ"א ואני שמעתי עוד בשם גדול אחד שאמר טעם נכון לזה כי בעת שעמדו על הר סיני וקבלו התורה [כי בעשרת הדברות נתגלה להם עי"ז כל חלקי התורה כמו שכתב רב סעדיה גאון שבעשרת הדברות כלולה כל התורה] וירדו מן ההר לביתם לא מצאו מה לאכול תיכף כ"א מאכלי חלב כי לבשר צריך הכנה רבה לשחוט בסכין בדוק כאשר צוה ה' ולנקר חוטי החלב והדם ולהדיח ולמלוח ולבשל בכלים חדשים כי הכלים שהיו להם מקודם שבישלו בהם באותו מעל"ע נאסרו להם ע"כ בחרו להם לפי שעה מאכלי חלב ואנו עושין זכר לזה:
(12) Dairy foods - See Magen Avraham and I also heard in the name of a Gadol who said a correct reason for this: that when Bnei Yisrael stood at Har Sinai and accepted the Torah(When the 10 commandments were revealed, and through this all the parts of the Torah were also revealed to them, like that Rav Sadya Gaon writes, that the 10 commandments sustain the whole torah) and they went down from the mountain to their homes they found nothing to eat immediately except for dairy foods because for meat they would need much preparation to slaughter with a checked knife like Hashem commanded... And to clean out the strings of forbidden fats and blood, and to rinse and salt(the meat). Also to cook in new vessels because their previous vessels had been used to cook meat (that they now discovered was actually trief) had been used in the last 24 hrs (which is what is forbidden Biblically) so their old vessels were forbidden to them, therefore they choose on account of the time to eat dairy foods and we do this as a commemoration of this.
(17) And I have said: I will bring you up out of the affliction of Egypt unto the land of the Canaanite, and the Hittite, and the Amorite, and the Perizzite, and the Hivite, and the Jebusite, unto a land flowing with milk and honey.
(7) Thou art all fair, my love; And there is no spot in thee. (8) Come with me from Lebanon, my bride, With me from Lebanon; Look from the top of Amana, From the top of Senir and Hermon, From the lions’dens, From the mountains of the leopards. (9) Thou hast ravished my heart, my sister, my bride; Thou hast ravished my heart with one of thine eyes, With one bead of thy necklace. (10) How fair is thy love, my sister, my bride! How much better is thy love than wine! And the smell of thine ointments than all manner of spices! (11) Thy lips, O my bride, drop honey— Honey and milk are under thy tongue; And the smell of thy garments is like the smell of Lebanon. (12) A garden shut up is my sister, my bride; A spring shut up, a fountain sealed.
Reason 6: The Magic Number
Rav Menachem Mendel of Ropshitz
The gematria (numerical value) of the Hebrew word for milk, chalav, is 40. We eat dairy foods on Shavuot to commemorate the 40 days that Moses spent on Mount Sinai receiving instruction in the entire Torah. Moses spent an additional 40 days on Sinai, praying for forgiveness following the Golden Calf, and then a third set of 40 days before returning with a new set of stone tablets. The numerical value of chalav, 40, has further significance in that there were 40 generations from Moses who recorded the Written Torah, till the generation of Ravina and Rav Ashi who wrote the final version of the Oral Torah, the Talmud.
Rabbi Dovid Meisels
Further, the gematria of gevina (cheese) is 70, corresponding to the "70 faces of Torah."
גבינה = 3 + 2 + 10 +50 + 5 "gevina"
(ז) וּפָרָ֤ה וָדֹב֙ תִּרְעֶ֔ינָה יַחְדָּ֖ו יִרְבְּצ֣וּ יַלְדֵיהֶ֑ן וְאַרְיֵ֖ה כַּבָּקָ֥ר יֹֽאכַל־תֶּֽבֶן׃
(7) And the cow and the bear feed; Their young ones shall lie down together; And the lion shall eat straw like the ox.
Rabbi Irving Greenberg, The Jewish Way: Living the Holidays
Just as Passover evokes the future universal redemption, so Shavuot evokes the future universal covenant. The prophet predicts that even carnivorous animals will turn vegetarian [...] hence follows the anticipatory dairy meal of Shavuot.
Reason 9: Breast milk and Menstruating!
(ו) חלב. יש הרבה טעמים ומ"כ הטעם דאיתא בזוהר שאותן ז' שבועות היו לישראל שבעה נקיים דוגמת אשה המיטהרת מנדתה וידוע שדם נעכר ונעשה חלב, והיינו מדין לרחמים, ומנהג אבותינו תורה היא אך יש ליזהר שלא יבא לידי איסור ועיין ביורה דעה סי' פ"ט דא"צ להפסיק בברכת המזון אם אינו אוכל גבינה קשה, ויזהר ליקח מפה אחרת:
(6) There are many reasons for eating dairy, I will copy the reason written in the Zohar: the previous seven weeks were in parrallel to the seven clean days a woman counts before immersing in the mikvah, we know that blood becomes milk, this is parallel to the movement from judgement to mercy, and the practice of our fathers is in our hands to fulfill. However, it is important to be careful not to violate any prohibitions (see Yoreh Deah 89)...
...ויעלת חן שמעלת חן על לומדיה דדיה ירווך בכל עת למה נמשלו דברי תורה כדד מה דד זה כל זמן שהתינוק ממשמש בו מוצא בו חלב אף דברי תורה כל זמן שאדם הוגה בהן מוצא בהן טעם באהבתה תשגה תמיד
...That the Torah bestows grace upon those who study it. “Let her breasts satisfy you at all times”; why were matters of Torah compared to a breast? Just as with a breast, whenever a baby searches it for milk to suckle, he finds milk in it, so too, with matters of Torah. Whenever a person meditates upon them, he finds new meaning in them.