It is customary to blow Shofar during the month of Elul. The reason for this is that since the sound of the Shofar causes one to tremble, it thus urges people to repent before Rosh Hashanah (טור אורח חיים סימן תקפ"א בשם פרקי דרבי אליעזר). As the pasuk says "אם יתקע שופר בעיר, והעם לא יחרדו" could it be that the Shofar will be sounded in a city and the people will not tremble?! (עמוס ג', ו').
Perhaps this could be understood in the following perspective. The Navi exclaims; "אריה שאג מי לא יירא", when the lion roars, who does not shudder in fear? (עמוס ג', ח') The קדמונים interpret the word אריה as reference to אלול, ראש השנה, יום כיפור, הושענא רבא, the four prime periods of Heavenly Judgement. Accordingly, the Navi is 'telling' us to be fearful of the Judgement.
The ישמח משה, explains this idea with an innovative twist. The Seforim Hakedoshim teach that there are two aspects of Yiras Shamayim. The simplistic level is יראת העונש, refraining from sin due to fear of the subsequent punishment. The higher degree is יראת הרוממות, withholding one's desire to go against the Torah out of trepidation for Hashem's Exaltedness. The Yismach Moshe quotes the words of Chazal that "אין ארי נוהם מתוך קופה של תבן אלא מתוך קופה של בשר", a lion only roars after eating meat (ברכות דף ל"ב.). Thus, since the lion is satisfied, the fear one experiences from the roar of a lion is not one of dread, because he is afraid of being attacked. Rather, it is one of awe and appreciation of the lion's grandeur (ישמח משה הפטרת וישב בד"ה הישאג).
We could perhaps apply the same principal with regard to our Avodah during this time of year. The month of Elul is an auspicious time for one to draw close to Hashem and, subsequently, acknowledge His Exaltedness and Glory. Hence, one should not only do Teshuvah out of fear of Judgement of Rosh Hashanah, where each deed is examined and analyzed in great detail. Preferably, one should also strive to attain the more desirable level which is יראת הרוממות, seeking to repent out of reverence of Hashem's Splendor.
Perhaps this is alluded to in the words of the Beis Yosef, who explains that upon hearing the sound of the שופר, one is filled with a reverent awe of Hashem's Glory and Magnificence (בית יוסף או"ח סימן תקפ"א).