Rav Saadia Gaon writes that there are 10 reasons why Hashem commanded us to blow the shofar on Rosh Hashana.
1. This day is the beginning of creation when G-d createad the world and ruled over it. At the beginning of a king's reign, they sounds the trumpets and shofars, to promulgate in all locations that the new reign has begun; likewise, we accept upon ourselves G-d's reign on this day [by sounding the shofar]...
2. RH is the 1st of the 10 Days of Teshuva, and we blow the shofar in order to announce that all those who wish to do teshuva, should do so!
3. The shofar reminds us of the shofar at Mt. Sinai, when we accepted the Torah.
4. The shofar reminds us of the Prophets, whose voice exhorts us to improve our ways.
5. The Shofar reminds us of the destruction of the Holy Temple and the cries of war. When we hear this sound, we will beseech Hashem to rebuild the Beis HaMikdash!
6. The shofar (ram's horn) reminds us of the Binding of Isaac, who was ready to give his life for the sake of Heaven; likewise, we should also dedicate ourselves to sanctifying His name, and He will surely recall us for the good.
7. When we hear the sound of the shofar, we will tremble. This is the nature of the sound of the shofar, it causes one to tremble with trepidation.
8. The shofar reminds us of the ultimate Day of Judgment (i.e. when Moshiach comes).
9. The shofar reminds us of the Gathering of the Exiles (when Moshiach arrives) when the shofar will be sounded, and this causes us to yearn for that time.
10. The shofar reminds us to believe in the resurrection of the dead.