The answer to this question is based on a law absolving servants from certain mitzvos. The Zohar teaches us that a gentile servant is not required to say kri’as sh’ma. This is because he is already beholden to his master. As long as he is beholden to his master, he cannot accept the yoke of another.
This concept applies to us as well. We are required to declare God’s Kingship over us on Rosh Hashana. But until we repent we are beholden to the evil inclination. As long as we are under the sway of the evil inclination we cannot accept God’s yoke. Just like the gentile servant, we must be first be free in order to subordinate ourselves to God.
This is the reason that the mazal for the month of Elul is besula/virgin. Just as the virgin is unattached, so too, must we free ourselves from all attachments in order to attach to God.
This concept also explains the juxtaposition of destroying Amalek at the very end of last week’s parsha, and the mitzvah of bikurim/bringing the first fruits at the beginning of this week’s parsha.
The mitzvah of bikurim is a physical expression of accepting the yoke of Heaven. We give the first of our produce to God and we make a specific declaration that we are doing this in order to show our subordination to Him. Amalek represents subordination to the evil inclination. The paragraph instructing us to destroy Amalek comes directly prior to the mitzvah of bikurim to teach us that in order to become beholden to God, we must first wipe out the Amalek that is within us.
This is the work that Chazal cut out for us during the month of Elul to enable us to accept God's Kingship on Rosh HaShana. May we merit it!