Many reasons have been given to the minhag Yisrael to eat milchigs on the Yom Tov of Shavuos. This D'var Torah is coming to suggest a new reason. The Medrash at the beginning of Matan Torah tells us of the scenario when Moshe Rabbeinu ascended Har Sinai to receive the Torah. As Moshe reached the top of the mountain the malachim began to be mekatraig denounce Moshe Rabbeinu who they were about to harm. It was then that Hashem transformed the countenance of Moshe to be that of Avraham Avinu. He addressed the malachim with astonishment How dare you prosecute this individual. Didn't you eat at the meal of this human being who is standing in front of us? Aren't you therefore embarrassed about your extreme belligerent unwelcoming behavior to the same individual who was incredibly hospitable to you?
It was then that their argument stance disappeared and Hashem said to Moshe that it was Avraham's merit that Beni Yisrael received the Torah. This is what the passuk conveys לקחת מתנות באדם to take the Torah which is called a gift, באדם in the merit of Avraham whom the passuk calls האדם הגדול בענקיםthe great man amongst giants.
One would initially think that Hashem's chastisement to the malachim was triggered due to their lack of middas Hakaras Hatov and ingratitude to Avraham. However we can say that Hashem's mussar to the malachim was ignited from a different place.
עולם חסד יבנה the world was built and continues to exists because of the middah of chesed. Hashem's characteristic of being a pure benefactor, a מטיב, to overflow His goodness to us was the motivation behind creating the world. He created the world with blueprints called the Torah. Therefore the Torah is called רחמנא from the word רחם which means compassion and love (in Aramaic). Rabbe Simlaee says the Torah is chesed because it begins with chesed and ends with chesed. After Adam and Chava ate from the Aitz Hadaas and they were embarrassed from their nakedness, Hashem made for them clothing מלביש ערומים. At the end of the Torah which deals with Moshe's demise it was Hashem who performed the chesed of burying Moshe Rabbeinu. If we find that the beginning and end of the Sefer Torah talks about acts of chesed we can rest assured that the middle contents also contain chesed. The Vilna Gaon in his letter to his wife and mother wrote כי רוב מצוות התורה לשמח את הברואים the majority of mitzvos in the Torah is to bring happiness to Hashem's creation.
The malachim could not fathom that Torah whose essence is chesed should be given over to Man to fulfill. Even though Man possesses a soul but he also has a body made from earth which causes one to gravitate towards physical and bodily pleasures which will naturally lead him to egocentricity, selfishness, self-centeredness, and narcissim if not controlled. The yetzer hara that man possesses is thirteen years older than his yetzer tov. His body is fueled and runs with the energy derived from physical consumptions. Man is therefore doomed not to become an emulation of His creator upon whom it is written חסד אל כל היום chesed without any side interests.
Hashem then said to the malachim You don't know my Children. Didn't you eat by Avraham in his tent oasis? You soon realized that he invited you in on the third day of his Milah when the pain increases to its height. It was also on that day that Hashem removed the shield of the sun and the weather was unbearable for a human to step outside his tent. Yet Avraham ran to you in this severe heat in order to invite you into his abode. He treated you like kings bringing to all of you the delicacy of eating individual tongues after three animals were slaughtered and prepared with fresh baked bread and mustard relish. One would have to pay an exorbitant price to receive such a lavish and exquisite meal which he offered to you without any charge as all. And you have the audacity to claim that the Torah of Chesed is not fit for people like Avraham!
The passuk says אלה תולדות השמים והארץ בהברהם. The translation is "these are the chronology of the skies and the earth when they were created. However Chazal see in its letters another word באברהם with Avraham. The Torah is conveying that the world was built through Avraham and his midda of chesed. The angels' existence today are only because of Avraham's kindness and selflessness. And you have the chutzpah to bad mouth My nation that they will be takers instead of givers?
On Shevuos we read Megillas Rus. Chazal say a similar description about the Megilla as they said above about Torah. Megillas Rus begins with chesed of how Rus the Moavite lost her husband and father in law, her possessions, esteem, and yet she followed her mother in law Naomi in her return to Eretz Yisrael, fulfilling all her instructions and wishes even though she was a complete stranger. The end of the Megilla talks about the chesed of how Boaz who was old did the mitzvah of Yibum with Rus even though her halachic eligibility of marrying a Yid was questionable. This also includes the chesed of Rus who sought out to marry a relative who would keep the memory of her departed husband alive. If the beginning of the Megillah was chesed and the end was chesed then the entire Megillah is replete with acts of chesed and לפנים משורת הדין.
This connection between the Chesed presented in the Torah and Megillas Rus is alluded to in the letters of תורה and רות for they both consist of the letters of תור. This is one of the reasons brought why we read Rus on Shvuos because the both shared the contents of a chesed life to be followed in the Torah and her own life of extraordinary chesed.
We also see in the Megilla the punishment of those who refuse to do chesed to others because of selfishness. Elimelech was rich but he possessed a צרת עין stingy and selfish. When a famine came to Eretz Yisrael he picked up and left in order to avoid the necessity to support and share his wealth with the poor people. His punishment was that he went to Moav who were also known for their stinginess and anti chesed hashkafa. Even if a Moav man converted he is still forbidden to marry into Yisrael for they refused to greet them with food and water to Yisrael when they left Mitzrayim. This is why Elimelech was punished that his children married Moavites because "takers" marry takers and "givers" marry givers.
Now we can answer why the custom is to have milchigs. The Kabbalists say that the color white corresponds to the middah of chesed. The Torah is full of chesed, Megillas Rus is full of chesed. By receiving the Torah the creation of the world became solidified as Chazal say on the words יום הששי that creation was hanging in limbo until the day Am Yisrael received the Torah. On Shavuos we experienced עולם חסד יבנה white. Therefore we eat white food products to remind us that Hashem's chesed to us is constant and we must emulate His ways and make Am Yisrael a happy nation as the Gaon wrote.