This list of reasons for blowing the Shofar on Rosh Hashanah was compiled by the 10th century sage, Rabbi Saadia Gaon.
The Shofar is like the trumpet which announces the coronation of a king. That is why it is used on Rosh Hashanah, the birthday of the universe. At that time, we accept Hashem’s Rulership. Our prayer and shofar blasts are like the coronation ceremony in which the assembly of Yisroel crowns Hashem as Sovereign.
Psalm 98
Sing to the LORD a new song, for he has done marvelous things; his right hand and his holy arm have worked salvation for him. The LORD has made his salvation known and revealed his righteousness to the nations.
He has remembered his love and his faithfulness to the house of Israel; all the ends of the earth have seen the salvation of our God. Shout for joy to the LORD, all the earth, burst into jubilant song with music; make music to the LORD with the harp, with the harp and the sound of singing,
(6) with trumpets and the blast of the ram’s horn– shout for joy before the LORD, the King… (9) let them sing before the LORD, for he comes to judge the earth. He will judge the world in righteousness and the peoples with equity.

The Shofar reminds us of the Shofar which blew when the Torah was given at Har Sinai; thus we are reminded to study and cherish Hashem’s Word.
Exodus 19:18-19 Now Mount Sinai was all in smoke because the LORD descended upon it in fire; and its smoke ascended like the smoke of a furnace, and the whole mountain quaked violently. When the sound of the trumpet grew louder and louder, Moses spoke and God answered him with thunder.
The Shofar reminds us of the voice of the Prophets, whose voices rang out like a Shofar blast in calling the people to do justice and mercy and follow Holy ways.
Ezek 33:1-6 And the word of the LORD came to me, saying, “Son of man, speak to the sons of your people and say to them, ‘If I bring a sword upon a land, and the people of the land take one man from among them and make him their watchman, and he sees the sword coming upon the land and blows on the trumpet and warns the people, then he who hears the sound of the trumpet and does not take warning, and a sword comes and takes him away, his blood will be on his own head. He heard the sound of the trumpet but did not take warning; his blood will be on himself. But had he taken warning, he would have delivered his life. But if the watchman sees the sword coming and does not blow the trumpet and the people are not warned, and a sword comes and takes a person from them, he is taken away in his iniquity; but his blood I will require from the watchman’s hand.’
The Shofar, since it is a ram’s horn, reminds us of the binding of Yitzchok, when the Lord provided a ram to be sacrificed instead. Thus we are called upon to be as faithful to Hashem as Avraham, and be inspired by his example of sacrifice and love of Hashem.
The Shofar calls us to be humble- its mighty blast reminds us of the mightiness of Hashem and the fact that Hashem is everywhere at all times.
On the Day of Judgment, a Shofar will be blown to announce Hashem’s Rulership- our Shofar blasts remind us to prepare for Hashem’s examination of our deeds.
The Shofar foreshadows the jubilant return to freedom and peace when we all end up in Yerushalayim in the time of Moshiach- it reminds us to have hope and faith in Hashem’s saving power.
The Shofar will be blown in Messianic times to announce the redemption of the whole world, when all nations will recognize that Hashem is One.
The Shofar calls us to examine our deeds and return to Hashem, who will always accept us if we are sincere. This is why we blow the Shofar on Rosh Hashana which is the first of the Ten Days of Awe.