To begin with, we blow the shofar on Rosh Hashana because God said so. See Numbers 29:1.
וּבַחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ מִקְרָא קֹדֶשׁ יִהְיֶה לָכֶם כָּל מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ יוֹם תְּרוּעָה יִהְיֶה לָכֶם
And in the seventh month, on the first day, there shall be a holy convocation for you; you shall not perform any mundane work. It shall be a day of shofar sounding for you.
Our tradition assigns a great deal of symbolism to the shofar blowing. Two major themes are summarized very nicely by R' Joseph B. Soloveitchik, as I found quoted in Machzor Mesoras Harav (p. 440).
The mitzvah of shofar itself contains two mutually exclusive motifs. On the one hand, the blowing of the shofar heralds God's entrance and when we find ourselves in His Presence, there is an obligation to engage in Shira, song. On the other hand, the shofar also reflects Tze'aka, crying out. According to the Ramban (Milchamos Hashem to Rosh Hashanah, 11a in Rif), because we are judged on Rosh Hashanah, the holiday contains the motif of a fast day, when Tze'aka is an integral compoent. The dialectic of the shofar reflects the dialectic of Rosh Hashanah itself. The Sefer HaChinuch (Mitzvah 311) indicates that although man should be frightened on this Day of Judgement, the very fact that man can indeed be judged is a source of pride for him. Animals cannot be judged; they are not respondible for their actions. If man is capable of being judged, it is because he is a responsible being, the choicest of God's creations, who can attain atonement for his sins (Noraot HaRav, Volume 1, pp. 77-80).
If you're interested in more details about the symbolism of the Shofar, Rav Sa'adia Gaon listed ten specific allusions in the shofar:
- It commemorates the creation of the world and the "coronation" of Hashem.
- It announces the beginning of the amnesty period that is the Ten Days of Repentance.
- It reminds us of the revelation at Sinai.
- It reminds us of the admonitions of the prophets and their calls to repentance.
- It reminds us of the Temple, where shofars were sounded.
- It reminds us of Avraham's faith at the binding of Yitzchak.
- Its sound inspires fear in the hearts of those who hear it.
- It reminds us of the day of judgement to come.
- It inspires us to yearn for the ingathering of the exiles.
- It recalls the resurrection of the dead.