Why do most Ashkenazim read Megillas Rus on Shavuos?
I am sure there are multiple answers to this question. I would like as many as possible, with a preference for answers from earlier sources.
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מה ענין רות אצל עצרת, שנקראת בזמן מתן תורה? ללמדך שלא נתנה תורה, אלא על ידי יסורין ועוני וכו'
What is the connection between Ruth and Atzeret, so that it is read on Atzeret, the time of the giving of the Torah? To teach you that the Torah is only given through suffering and poverty.
This is in turn comes from the beginning of Ruth Zuta, an earlier midrash. However, the most practical reason is that the story takes place around wheat harvest (Ruth 2:23 and Machzor Vitry), and Shavuot was the festival of this agricultural event.
Many other interesting things are listed on this Chabad site:
The last verses show (Ruth 4:17), how King David descended from Ruth. Yerushalmi Chagigah 2:3 says in the gemara of the third halakhah that he died on Shavuot (א"ר יוסי בי רבי בון דוד מת בעצרת), and we know from the case of Moses that the most righteous people's years are completed, and they die on the day, when they were born (Rosh Hashanah 11a, Kiddushin 38a). So we read the Book of Ruth in King David's honour (Bekhor Shor to Bava Batra 14b).
The numerical value of Ruth's name is 606 (תר"ו). This alludes to the 606 additional laws that we received at Sinai in addition to the 7 Noachide laws. Ruth accepted upon herself these mitzvot as well when she converted (Birkei Yosef 494:11, see again Bava Batra 14b for a different explanation).
Ruth's case shows us, how important is the Oral Law that we also received on Shavuot. From Devarim 23:4 we may mistakenly infer that Amonites and Moabites can't marry Jews at all. However, Yevamot 69a makes it clear that it only refers to males and not to women (דאמר מר עמוני ולא עמונית מואבי ולא מואבית).
Although not everyone necessarily accepts teachings based on juxtaposition of verses (Berakhot 21b), it is quite thoughtful that after discussing the laws of Shavuot (Vayikra 23:15–21), the Torah tells us the laws of peah (Vayikra 23:22), a mitzvah that was fulfilled eminently by Boaz (Levush Hachur 494:2).
Sources again from Zvi Ron's article
In this article Zvi Ron makes an interesting suggestion, that the reading of Megillat Rut may not be directly linked to Shavuot itself, but instead to the weekly Torah readings that are read on the shabbatot around Shavuot:
Ruth parallels verses and themes found in the Torah readings right before and right after Shavuot. For example, in Parashat Bemidbar: Numbers 3:4 and Ruth 1:5 (death of brothers), Numbers 4:18 and Ruth 4:10 (cutting off), in Nasso, Numbers 5:8 and Ruth 3:12 (a redeemer), Numbers 6:24 and Ruth 2:4 (“May God bless you”), Numbers 6:27 and Ruth 4:14 (a name in Israel), and in Beha'alotcha, the section in Numbers 10: 29-30, where Yitro returns home “to my land and to my family,” a reverse parallel of Ruth. This is particularly significant in light of the custom reported in Masechet Soferim that Ruth was read beginning the Saturday night before Shavuot.
Hat-tip to @DoubleAA for linking to this fascinating article in a comment on the question.