The main mitzvah of Rosh Hashanah is to hear the shofar. According to the Nesivos Sholom, a great Rabbi from the previous generation, the shofar represents the cry of the broken heart that is in every Jew. In Hebrew, each letter has a number. The numerical value of the word shofar has the same numerical value as the phrase “shivron lev,” broken heart.
Throughout the year, we do things that we wish we hadn’t done; actions that are not in line with our true will. Yet we are so numb today, with all of the distractions of the Google Age. It’s so hard for us to access that part of us that wants to our best selves, that wants to have a connection to G-d and achieve real meaning and happiness in our lives. In commanding us to hear the shofar on Rosh Hashanah, G-d gives us the opportunity to let out a cry from this part of ourselves, which is usually buried within us the rest of the year. This cry, which is like the loving cry of a child yearning for his parent, is so powerful that it awakens G-d’s love for us, and overrides His love for justice.
(Source: Rabbi S. Binyomin Ginsberg, Gems From the Nesivos Sholom, p. 149-156 (based on the Kuf Lamed/Kuf Lamed Bet of the second volume of the Nesivos Sholom)