Those that have the Minhag to put out greenery in the Shul on Shavuos, why do we do it?
Mishna B'rura 494:10 says it's to remind us of the joy of Sinay (Mount Sinai), around which stuff was growing. Taame Haminhagim 617 cites this from the L'vush.
Magen Avraham 494:5 gives a reason for trees specifically: to remind us to pray for fruit, on which judgment is passed on Shavuos.
The compilation Pardes Eli'ezer lists the following explanations:
Based on the same L'vush source cited by @msh210, Lechem Sh'lomo says the particular happiness felt upon receiving the Torah owed to their witnessing a desert mountain bloom with plants, which reminded them that even the barren wasteland of their lives of sin had the potential to turn positive again with the reception of the Torah.
Midrash Talpiyos says R'uven went to collect flowers on the eve of Shavu'os (both happen at the time of the wheat harvest), which resulted ultimately in the birth of Yisachar, whose unique trait is Torah study.
Chasam Sofer contends that we place fragrant plants around because the words of revealed Torah at Sinai made the world fragrant.
Perhaps it is because Torah represents growth. Am Yisrael is compared to a rose (Canticles 2:2), which is a type of plant, and the Zohar states that Yisrael ve'Oraita Chad Hu - Israel and Torah are a singular entity. Plants grow, and roses are included in plants. The fact that Israel is a plant means that the Torah is a plant. The Torah is also referred to as a tree - etz chayim hee, which is further proof to this. The Torah wants us to grow spiritually, and (IMHO) Ma'amad Har Sinai is a religious experience that we should associate with spiritual growth.