Chazal tell us, "Recite before Me [pesukim of] kingship so that you proclaim Me King over you. Recite before me [pesukim of] remembrance so that I will remember you favorably. And by what means? With the shofar" (Rosh Hashana 16b). While the beginning of the above Gemara seems to imply that with the recitation of the pesukim of malchiyos one has proclaimed Hashem King, from the conclusion it is clear that the mere recitation of the pesukim is not enough. Proper acceptance of Hashem's Kingship can only be achieved through the blowing of the shofar.
What does the shofar accomplish that the pesukim cannot? Rav Wolbe explains (Mamarei Yemei Ratzon p. 376) that the shofar is the instrument used for revelations. We say in the mussaf of Rosh Hashana, "You appeared to Your holy nation, with the sound of the shofar You revealed Yourself to them." The Rambam writes that the shofar blown each Rosh Hashana acts as a bugle to wake people from their reverie: "Wake up, sleeper, from your sleep, awaken from your slumber, do teshuva and return to Hashem." The shofar arouses a person and reveals to him exactly where he stands in his spirituality. Although the recitation of the pesukim is the acceptance of Hashem's Kingship, if this recognition has not penetrated the heart then the acceptance is not complete.
While there are many esoteric reasons behind the mitzvah of shofar, the Shelah explains us to the simple significance behind the various blasts. Each set of blasts begins with a tekiah. The straight sound of the tekiah symbolizes the straightness with which each man was created. Unfortunately, man often leaves the course Hashem charted for him and decides to follow his own path, thereby replacing straightness with crookedness. The shevarim symbolizes the groan of man when he realizes just what he has done and groans over his transgressions. The teruah, the short, stuttered wail, symbolizes the even greater feeling of bitterness resulting from truly comprehending one's forlorn state. He wails because he realizes that he simply cannot continue on his wayward path, and he makes a decision to return to the original state of straightness in which he was created - the final tekiah. This explanation complements the above idea: The shofar allows us to feel how far we have fallen due to our transgressions.
The problem is that we are completely numb and we simply do not feel! We behold tragedies, and are aware of our own sins and we remain totally unaffected. The Gemara tells us, "Look how terrible the dust (i.e. a fringe aveirah) of shemittah is. A person does business with fruit of shemittah, and is punished that he will end up sending his possessions. If he doesn't feel it (he continues his wayward actions) he will end up selling his real estate. If it does not come to his hand he will end up selling his house. Why does it say the first time [referring to his continuing to sin] 'if he doesn't feel it' and the second time it says 'if it doesn't come to his hand?' Because when a person transgresses an aveirah once and then a second time, it seems to him that the aveirah is permitted (and he no longer feels it)" (Arachin 30b). Rashi explains that "not feeling it" means that he was not shaken by the punishment to cause him to refrain from sinning.
Regesh - a heartfelt feeling [of remorse] - is what galvanizes a person to do teshuvah. What do we do if we have lost our regesh? Rav Yisrael Slanter instructs us, "The first step in avodas Hashem is regesh; a person should study a statement from Chazal and review it many many times until it makes an impression upon him and he feels what he is lacking. This will bring him to subdue and then completely overcome his yetzer hara, and it will bring him to a level where he is ecstatic and joyful to serve Hashem."
Choose one statement of Chazal that relates to an aspect of avodas Hashem which you would like to improve. Recite it again and again and take it with you into Rosh Hashana. When the shofar is blown, appreciate the message being conveyed: Hashem, I went awry in this area and I want to come back home and accept your Kingship wholeheartedly. Regesh leads to remorse, which leads to subjugating one's yetzer and makes way for an exhilarated desire to accept Hashem's Kinghip - the purpose of Rosh Hashana!