1. Since Rosh Hashana is the beginning of creation when G-d created the world and became King, so it is the custom among kings to blow the horns as a way to declare their kingship. This is one reason we blow the Shofar.
2. Since Rosh Hashana is the first day of the Ten Days of Repentance, we blow the Shofar on that day as a warning in the same way kings warn their people before executing a decree. This way they cannot complain later that they weren’t warned.
3. To remind us of the revelation at Mount Sinai, as it says: “The sound of the Shofar is very strong”, and we will accept on ourselves to first do then hear just like our fathers did at Mount Sinai.
4. To remind us of the words of our prophets who were compared to the sound of the Shofar: “And a listener hears the sound of the Shofar but does not take heed…” (Yechezkel 33)
5. To remind us of the destruction of the Temple, and the sounds of our enemies’ war horns, so that we will pray to G-d for the rebuilding of the Temple.
6. To remind us of the ram that was slaughtered instead of Yitzchak who sacrificed his life during the Akeida. So too, we must also sacrifice ourselves for the sake of G-d so that He will remember us for the best. Our sages have said: “Blow before Me a horn of a ram so I will remember the Binding of Yitzchak the son of Avraham for you and your memory will come before Me.”
7. So that the sounding of the Shofar will awaken us to be in awe of G-d: “Is the Shofar ever sounded in a city and the people not tremble?” (Amos 3)
8. To remind us of the great and awesome Day of Judgement, as it says: “The great day of Hashem is near, it is near and hastens greatly, a day of trumpet and battle.” (Tzephania 1)
9. To remind us of the day of the gathering of the exiles and our yearning for it: “It shall be on that day that a great Shofar will be blown, and those who are lost in the land of Assyria…will come together…” (Isaiah 27)
10. To remind us of the resurrection of the dead and to believe in it: “All you inhabitants of the world and dwellers of the earth, you will see when the banner is hoisted up upon the mountains, and when the Shofar sounds you will hear!” Rosh Hashana is called Yom Truah and not Yom T’kiah because of the word Shivron (brokenness). This alludes to our broken hearts, regret and repentance. We do not blow the Shofar on the first day of Rosh Hashana when it falls on Shabbat, rather we blow it on the second day.