Shavuos pei daled: Why We Read Rus on Shavuos

 to be refined later, putting down so it doesn't get lost.

Abudraham: We lein Rus on Shavuos. 

Gives two reasons. 

It was in the harvest season, as is Shavuos.

Rus was misgayer, as we were at Mattan Torah.

Both explanations are adequate because one assumes that "we decided to lein all the five Megillos on the five moadim, so any connection will suffice for pairing them up, for assigning them to a particular one."

Four more satisfying explanations, purely my judgment.

1. Birkei Yosef quoting Simlah Chadasha: Dovid died on Shavuos, so he must have been born on Shavuos. So we read Rus not so much because of Shavuos, but in honor of Dovid Hamelech. (Unless you can come up with something that associates the character of malchus beis Dovid with the essence of Shavuos.)

2. RTzPF Mikra'ei Kodesh Pesach III Miluim for Shavuos 35: תורה שבכתב without תורה שבעל פה is impossible and ruinous. From Torah shebichsav alone, we would never have accepted Rus into K'hal Hashem, and where would Beis Dovid come from? So the lesson is that at the Yomtov of Kabalas Hatorah, we must keep in mind that we could not exist with Torah shebichsav alone, and Torah shebaal peh is no less essential.

3. Pri Megadim in the Hakdama to OC, and Chidah intro to his sefer Simchas HaRegel: Rus had the courage and certainty and אמונה ובטחון  of לכתך אחרי במדבר בארץ לא זרוע, to enter into her covenant with the RBSO in the worst situation, poor, widowed and friendless in a strange land, similar to our conditions in the midbar when we left Mitzrayim. Davka because of that courage and certainty, that אמונה ובטחון, she lived to see her great-great-grandson seated on the throne of Israel and master of his world (Rashi Melachim I 2:19). 

4.  Based on the Elya Rabba Os yud: Shavuos is the yomtov of Kabalas Hatorah. Our kiyum hamitzvos is not meant to be limited to the letter of the law. The mitzvos are meant to be transformative. When it says in the Aseres Hadibros lo sirtzach, it includes מלבין פני חבירו. When it says לא תגנב it includes גניבת דעת. When it says לא תנאף  it includes of the זימה that the RBSO hates. This is why at the end of Parshas Yisro the Torah interposes three mitzvos-

  לא תעשון אתי אלהי כסף ואלהי זהב לא תעשו לכם 

מזבח אדמה תעשה־לי וזבחת עליו את־עלתיך ואת־שלמיך את־צאנך ואת־בקרך בכל־המקום אשר אזכיר את־שמי אבוא אליך וברכתיך 

ואם־מזבח אבנים תעשה־לי לא־תבנה אתהן גזית כי חרבך הנפת עליה ותחללה 

ולא־תעלה במעלת על־מזבחי אשר לא־תגלה ערותך עליו

Elohei chesef teaches that Avodah Zara should be so unthinkable that we do not even allow images that others have worshipped into our houses. The din of avnei gazis is because Retzicha is so horrible that the RBSO does not allow iron shaped stone on the mizbeiach. Ni'uf is so repulsive that the Mizbei'ach should have a ramp so that kohanim doing the avoda walk and comport themselves like malachim. 

This expansive definition of dinim expresses itself in explicit mitzvos, and in Avos and Toldos of mitzvos, and in the din of נבל ברשות התורה.

This is why the nations declined the invitation to Mattan Torah. They were already bound by the dinim of retzicha and gneiva and arayos, but that went only as far as the literal application, and Torah requires that we become a Torahdikkeh nation, ממלכת כהנים וגוי קדוש.

The only way that learning Torah can have this effect on a person is if he makes himself a beis kibbul for Torah. Only a person who is daveik bashem is such a beis kibbul. How can a person be daveik bashem? Only through והלכת בדרכיו. What does that entail? Chesed. מה הוא....

והא ראייה, from Sotah 14a, דרש ר' שמלאי: תורה – תחילתה גמילות חסדים וסופה גמילות חסדים 

Megillas Rus, from beginning to end, is a lesson of Chesed. This is very evident, so you don't need it, but the Malbim makes this a theme of the Megilla.

אמר ר' זעירא: מגילה זו אין בה לא טומאה ולא טהרה ולא איסור ולא היתר, ולמה נכתבה? ללמדך כמה שכר טוב גומלי חסדים 

So on Shavuos, when we are mekabeil mitzvos as we would be standing at Har Sinai, we have to make a kabalas mitzvos that will transform us into a ממלכת כהנים וגוי קדוש, not people who are נבלים ברשות התורה. The only way to do that is to be muchshor such that the words of Torah will penetrate our souls and transform us. That requires that we become like Rus, the quintessence of Chesed.