There is a beautiful parable that illuminates the reason for sounding the Shofar in the month of Elul and on Rosh Hashanah.
Two people came before a judge in a court to have their case heard. One of the litigants realizes that the judge is from his part of the country. He proceeds to present his case in the special dialect of the region, known only to the local people. The sound of the litigant’s words, spoken in the same rhythm and pitch as his parents, moves the judge. From that moment forth, the case is effectively over!
When the Jewish people come before G-d and speak in the language of the Shofar, they speak in a special language, known only to them and G-d. G-d’s heart is effectively swayed (so to speak). As we learn in Psalms, “Happy is the people that understand the call of Teruah of Hashem, in the light of Your presence they shall walk”.
It is known in Judaism that a horn made of a cow or ox may not be used as a Shofar. The Talmud explains that the reason for this is that “the prosecution cannot also act as the defense”. The horn of a cow serves as a reminder of the Golden Calf and can therefore not be used in Israel’s defense when they come to plead before G-d.
In fact, according to the Talmud, the horn of a ram is the most preferable kind of Shofar as it is explained than when G-d hears the call of the Shofar He is reminded of the binding of Isaac and the ram that was offered in his place and he will consider it as thought the Jewish people are offering themselves up before G-d. Ultimately, the ram’s horn is a reminder of Abraham’s total dedication to G-d and the Jewish people, feeling lowly and unworthy, present their forefather’s deeds before G-d, hoping that they will serve as protection for them.
The Biala Rebbe, may he live days that are pleasant and long, explains this passage in the Talmud as follows. The call of the Shofar moves every Jew to wanting to break down the walls between himself and G-d. Each Jew who hears the Shofar wishes to offer himself to G-d in complete self-sacrifice. Just as Isaac was willing to offer himself up to G-d, every Jew has an inner desire to do the same and the Shofar’s call exposes this desire.
Adapted from http://www.nishmas.org/minhagim/shofar.htm