There are numerous reasons which provide insights into this mitzvah. Here are a few, and I wish you a "Good and Sweet New Year"!
1) Rosh Hashana is the day that commemorates the creation of the world and it is described as the "coronation" of God as “King of the Universe”. As it is customary to sound a trumpet at a king's coronation so we blow the shofar on Rosh Hashana.
2) The ram's horn reminds us of the Binding of Isaac when Abraham demonstrated his absolute faith in God by being prepared to sacrifice his son. God demonstrated His absolute love for Abraham by having him sacrifice a ram in his place.
3) The shofar blast marks the beginning of a period of amnesty which is known as The Ten Days of Repentance. Repentance is based on the fact that since humanity has been given free-will, and our actions are not pre-determined, we must take responsibility for our actions.
4) When the Jews accepted the Torah at Mount Sinai, the sound of the shofar is described as "continuously increased and was very great" (Exodus 19:19). The shofar serves to remind us of the revelation at Mt. Sinai and therefore to renew our acceptance of Torah morality as absolute and God-given.
5) The shofar reminds us to pray for the rebuilding of the Temple where trumpets and shofars were sounded. Just as God manifests His presence in the world in specific places like the Temple in Jerusalem, He also manifests His presence at special times, such as the Ten Days of Repentance.
6) The sound inspires us to yearn for the ingathering of the exiles from all over the world, which will be heralded by the sound of a shofar. There will be absolute unity amongst the Jewish People and our connection to the Land of Israel will again be absolute.
