The month of Elul carries a structural role in preparing Israel for judgment: the Gemara already records that royal messengers were dispatched in Elul specifically "due to Rosh HaShanah" (Rosh HaShanah 18a), and Tehillim 81:4 provides the call to "sound the shofar at the new moon" that the Tur (Orach Chayim 581) reads as the scriptural warrant for the entire month's penitential practice — Rosh Chodesh Elul being the very day Moshe ascended for the second Tablets, making it an established time of Divine favor.
The name of the month is itself encoded with meaning: both the Shelah (Rosh HaShanah, Ner Mitzva 2) and the Kitzur Ba'al HaTurim (Devarim 30:6) note that the words "et levavecha ve'et levav" form the acronym aleph-lamed-vav-lamed, marking Elul as the month designated for repentance, and the Mishnah Berurah (581:1) adds the further mnemonic of "Ani le-dodi ve-dodi li" as an acrostic for Elul, reflecting the month's quality of closeness between Israel and the Holy One — a reading the Sefat Emet (Devarim, For the Month of Elul 4:2) develops by teaching that the fulfillment of "I am to my Beloved" during Elul itself brings about the response of "my Beloved is to me" in Tishrei.
The thirteen attributes of Divine mercy are understood to be awakened during Elul as a special season of favor: the Sefat Emet (Devarim, For the Month of Elul 1:3) teaches that in Elul, as on Shabbat, the thirteen sources of compassion are aroused, and the capacity to receive their influence depends on how fully a person nullifies self-will and acts with measure — matching the bounded channel through which infinite Divine mercy flows toward created beings.
The practical forms this penitential season takes — rising before dawn for Selichot from Rosh Chodesh Elul, sounding the shofar each morning, and reciting ten chapters of Tehillim daily in public — are described by the Mishnah Berurah (581:1), the Mishnah Berurah (581:3), and the Mishnah Berurah (581:2), who further records the widespread custom to recite Mizmor le-David (Tehillim 27) morning and evening from Rosh Chodesh Elul through Shemini Atzeret.
The Rambam (Hilchot Teshuvah 2:6) establishes that while repentance and prayer are fitting at all times, during the ten days between Rosh HaShanah and Yom Kippur they are especially fitting and immediately accepted — citing Yeshayahu 55:7's call to "seek Hashem when He is to be found" — a principle that the Shelah (Rosh HaShanah, Ner Mitzva 2) applies retroactively to Elul itself as the final month of the year, the logical time to repair what was damaged so as not to enter the Day of Judgment still stained.
Elul is also a season for the moral accounting described by the Rambam (Hilchot Teshuvah 3:4), who teaches that a person should regard himself throughout the year as if his merits and sins are evenly balanced — so that each single sin or single mitzvah tilts the entire world — and Akavya ben Mahalalel (Pirkei Avot 3:1) grounds this reckoning in three contemplations: one's origin, one's destination, and that one will give account before the King of Melachim.
Likkutei Sichos (Ki Teitzei) concludes, on our reading of the cited page, that the twelve days from Chai Elul through Rosh HaShanah function as a structured accounting corresponding to each of the year's months, and that this spiritual reckoning must generate concrete positive resolutions — for the spiritual awareness gained inwardly must express itself in actual, tangible change in daily life.