First of all, Rema adds in [Orach Chayim 490:9][1] that we read Ruth on Shavuot.

The most practical reason is that the story takes place during wheat harvest ([Ruth 2:23][2]), and Shavuot was the festival of this agricultural event.

However, many other interesting things are listed on this [Chabad site][3]:

- The last verses show, how King David descended from Ruth. [Yerushalmi Chagigah 2:3][4] 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][5] that the most righteous people's years are completed, and they die on the day, when they were born ([Rosh Hashanah 11a][6], [Kiddushin 38a][7]). So we read the Book of Ruth in King David's honour.

- 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.

- Ruth's case shows us, how important is the Oral Law that we also received on Shavuot. From [Devarim 23:4][8] we may mistakenly infer that Amonites and Moabites can't marry Jews at all. However, [Yevamot 69a][9] makes it clear that it only refers to males and not women (דאמר מר עמוני ולא עמונית מואבי ולא מואבית).


  [1]: https://he.m.wikisource.org/wiki/%D7%A9%D7%95%D7%9C%D7%97%D7%9F_%D7%A2%D7%A8%D7%95%D7%9A_%D7%90%D7%95%D7%A8%D7%97_%D7%97%D7%99%D7%99%D7%9D_%D7%AA%D7%A6_%D7%98
  [2]: http://mechon-mamre.org/p/pt/pt2902.htm#23
  [3]: https://www.chabad.org/library/article_cdo/aid/2836850/jewish/Reading-Megillat-Ruth-on-Shavuot.htm
  [4]: http://mechon-mamre.org/b/r/r2b02.htm
  [5]: https://judaism.stackexchange.com/a/88787/15256
  [6]: http://hebrewbooks.org/shas.aspx?mesechta=9&daf=11&format=pdf
  [7]: http://hebrewbooks.org/shas.aspx?mesechta=20&daf=38&format=pdf
  [8]: http://mechon-mamre.org/p/pt/pt0523.htm#4
  [9]: http://hebrewbooks.org/shas.aspx?mesechta=14&daf=69&format=pdf