Among the ten reasons presented by Rav Sa’adia Gaon for blowing the Shofar is that the Jewish Shofar symbolizes G-d’s coronation over the world. In the same way that trumpets are blown at a coronation, so too the Jewish people blow the Shofar on Rosh Hashanah, the Jewish New Year and the day on which we proclaim G-d’s Kingship.
On Rosh Hashanah, the morning prayers are introduced with the cantor singing the word “Melech”, meaning “King”. The central theme of Rosh Hashanah is just that- Kingship- the Kingship of G-d.
On Rosh Hashanah, the entire world gets judged on their actions from the previous year and it is decided by G-d how their coming year will be. Therefore, in the month leading up to Rosh Hashanah, there is much emphasis in Jewish circles put on repentance before G-d. the pinnacle of this repentance is on Rosh Hashanah, when the Jewish people spend an intense two days praying, hearing the Shofar being blown and reflecting on their behavior in the past year.
Rav Yitzchok Blazer, a disciple of Rav Yisroel Salanter explained that proper repentance is dependent on realizing the severity of one’s sin and this cannot be achieved without acknowledging that G-d is our Melech- our King. In the same way that one will not feel obliged to go according to the laws of a country without recognizing the power of the ruling authorities, so too Jewish people will not feel obliged to live according to G-d’s laws without first accepting His power. One cannot truly return to G-d without understanding that G-d holds him responsible for his actions.
Rav Avigdor Miller writes that the Shofar arouses people to repent. It isn’t enough to just think of repentance in one’s heart. That is the beginning, and a good one at that. But it needs to go further than that. The Shofar is an effective way of sounding this desire to return to G-d. In the same way that when praying in Judaism, there is a need to actually pronounce the words one is reading and not just to mouth them in order to make praying effective, so too we need to hear the voice of the Shofar and not leave the thoughts of repentance as nothing more than that.
Every year on Rosh Hashanah we are renewing our loyalty to our G-d through the blowing of the Shofar. We recognize that G-d is the King since the time of Creation and continues to be King now and will be so for eternity. It is relayed that the Gra would be full of joy at the time of Shofar blowing because he explained that just as a nation is full of joy when crowning their King, so should we be when coronating G-d over the universe.