The central commandment of Rosh Hashana, and the defining feature of this festival in the Torah, is the sounding of the shofar. There is also a custom to sound the shofar each morning for the month preceding Rosh Hashana as you write. I will try to explain the reason for this custom.
Let me begin with a question: Why do we blow a shofar on Rosh Hashana? Rabbi Moshe ben Maimon (Rambam, Maimonides) writes: Even though the blowing of the shofar is a decree of the Torah, there is nevertheless a hint within it. That is, "Wake up... from your sleep... Search through your actions, return in repentance and remember your Creator... Look into your souls, improve your ways... and abandon evil..."
The basic reason for the commandment of shofar is to serve as a piercing “wake-up call”. A call to wake up and begin an accounting of our lives, become aware of our responsibilities and make positive commitments for the future.
In light of this explanation, we can understand the custom to begin blowing the shofar in the Synagogue a month before Rosh Hashana. We begin the process of “change” well before Rosh Hashana to avoid a last-minute, panicked rush. During the month before Rosh Hashana, known as Elul, the shofar is blown every day after the weekday morning services in order to wake us up from our apathy gradually, in the hope that we will not press the “snooze button” on our lives.