Elul: freeing oneself to accept Hashem's kingship

חודש אלול שניתקן לתשובה קודם ר"ה שהיא קבלת מלכותו יתברך כמ"ש אמרו לפני מלכיות.

The month of Elul, which was established for teshuvah before Rosh Hashanah — which is the acceptance of His kingship, may He be blessed, as it says, "Recite before Me [verses of] Malchuyos."

Elul was set as a time of teshuvah in preparation for Rosh Hashanah, the day on which we crown Hashem as King, as expressed in the Malchuyos verses of the davening.

ואיתא דמאן דכפות באחרא א"י לקבל עליו עומ"ש לכן צריכין בחודש הזה לצאת משיעבוד הסט"א ויצה"ר להיות נקי לקבל עומ"ש [לכן נקרא מזל חודש הזה בתולה] וזה ג"כ הקדמת מחיות עמלק קודם מצות ביכורים שהיא קבלת עומ"ש:

And it is brought that one who is bound to another cannot take upon himself the yoke of malchus Shomayim; therefore in this month we must go out from the servitude of the sitra achra and the yetzer hara, to be clean to accept the ol malchus Shomayim — [therefore the mazal of this month is called Besulah (Virgin)] — and this is also the placing of the wiping out of Amalek before the mitzvah of bikkurim, which is the acceptance of the yoke of Heaven.

One who is still enslaved to another master cannot truly accept the ol malchus Shomayim (yoke of Heaven's kingship); so during Elul a person must free himself from the grip of the sitra achra (the other side) and the yetzer hara, becoming pure and unattached. This is hinted by the month's mazal, Besulah — the untouched, pure maiden — and parallels the command to wipe out Amalek being placed before the mitzvah of bikkurim, since one must first remove the opposing power before he can fully accept Hashem's kingship.

Summary: Elul is the month of teshuvah that prepares us to crown Hashem on Rosh Hashanah. Since one who is bound to another master cannot truly accept the yoke of Heaven, the avodah of Elul is to break free from the servitude of the yetzer hara and the sitra achra and become pure — hinted by the month's mazal of the Besulah and by the Torah placing the wiping out of Amalek before the acceptance of Hashem's kingship in bikkurim.