One explanation is that Ruth's tremendous self-sacrifice in abandoning her country, her wealth and her comfort in order to become part of the Jewish people is a lesson to us all about priorities in life.
Another reason is that on Shavuot we all look upon ourselves as converts to Judaism. When the Jewish people came to Mt. Sinai, they unconditionally accepted the entire Torah and God's sovereignty. They purified themselves, the men were circumcised and the entire people entered into a covenant with God. Each of these steps is repeated by every convert to Judaism.
A third reason is that Ruth's great-grandson was King David, who was born on Shavuot (and 70 years later, died on Shavuot). The Messiah will ultimately trace his lineage to King David.