The most important Mitzva of Rosh haShana is listening to the Shofar.
The Tora does not comment explicitly the reasons for the Shofar, but our Rabbis did.
1. The Shofar was used in ancient Israel for the King's coronation. In Rosh haShana we state that God is our King ('ol malkhut shamayim), we are His subjects and as such we follow His rules. The King had the power to sentence or spare someone's life. In Rosh haShana we realize that our lives are in the hands of God, the supreme King and Judge. The key added word which is emphasized in the prayers of Rosh HaShana and Yamim Noraim is melekh. God is the King and Supreme Judge.
2. The Shofar also reminds us of aqedat Yitzchaq. Abraham Abinu was ready to sacrifice his own son to follow God's commandment. At the end, God told him to hold back because the test was over. Abraham saw a ram caught by its horns in a thicket and he offered it as a sacrifice to Him.
3. The Shofar was also used as a siren, alerting people the enemy is about to attack and their lives were in danger. In Rosh haShana the Shofar serves as a spiritual alarm clock inviting us to introspect on the fragility of our existence and repent. The Shofar is a wake up call urging us to plead with haShem and request His mercy when judging us.
In Hilkhot Teshuba 3:4, Maimonides explains that the sounding of the Shofar conveys the following message: "Wake up sleepy ones and those who slumber, arise. Inspect your deeds, remember your Creator. Those who forget the truth because of the temporary distractions and throughout the entire year devote their energies to the [pursue] of vanity and material emptiness which will not benefit , not save. Examine your ways and your deeds and abandon the paths of evil and the destructive [habits and] thoughts."
WATCH
