The shofar's call on Rosh Hashanah is fundamentally a summons to return: Mishneh Torah, Repentance 3:4 teaches that although the shofar-blast is a Scriptural decree, it carries an inner meaning — "Awaken, you sleepers, from your slumber; rouse yourselves, you slumberers, from your deep sleep" — calling each person to examine their deeds, return in repentance, and remember their Creator, while מנורת המאור, נר ה כלל ב חלק א פרק ד adds that because a person is immersed in desires throughout the year and does not rouse himself to repentance, the Torah commanded the shofar-blast on Rosh Hashanah precisely to awaken and remind Israel — for God desires not the destruction of the wicked but that they return and live.
This awakening resonates with the voices of the prophets: Yeshayahu 55:7 calls the wicked to abandon their ways and return to Hashem, who will have mercy and abundantly forgive, and Hoshea 14:3 urges Israel to take words and return to Hashem, offering the speech of their lips in place of sacrifices — together forming the scriptural backbone of the season's teshuvah imperative, while Joel 2:12-17 portrays the shofar in Zion as the signal to solemnize a fast and gather the entire people in assembly before God.
The Gemara in Rosh Hashanah 16a anchors this liturgically, teaching that the Holy One instructed Israel to recite Malkhuyot, Zikhronot, and Shofarot on Rosh Hashanah — Malkhuyot to crown Him as King, Zikhronot so that Israel's remembrance rises before Him for good, and the memory rises specifically through the shofar — so that the shofar-blast is the instrument through which Israel's standing before the Divine throne is renewed.
The Hasidic masters read the shofar as awakening a hidden inner voice: Sefat Emet, Devarim, Shabbat Shuva 21:4 teaches that the shofar-blast on Rosh Hashanah arouses the very voice heard at Sinai — the voice of the Beloved knocking — to awaken the heart to teshuvah, for even when Israel sleeps in its observance, a concealed point of "Anochi" remains buried in every Jewish soul, while Sefat Emet, Devarim, Rosh HaShanah 13:7 adds that the shofar alludes to the receiving of the Torah and demonstrates that the essential will behind living is to fulfill Torah — and this is the core of teshuvah: renewing our days as of old.
Likutei Moharan, Part II 8:14 interprets the verse "Sound the shofar at the moon's renewal" so that the blast (tik'u) alludes to the revelation of God's glory — the concept of affixing a peg in a secure place that becomes a throne of glory — while "bachodesh" alludes to the renewal (chidush) of the world through a new song, and "shofar" alludes to the prophetic "raise your voice like a shofar," weaving together the ideas of glory, renewal, and proclamation into a single Rosh Hashanah teaching.
Resisei Layla 51:1 develops the dynamic of awakening from below drawing forth response from above: in this world all our work is through arousal from below, and by awakening the matter of the future shofar-blast — the ultimate ingathering — we call out "Return us to You and we shall return"; for in truth everything is through judgment, and a gentle arousal from above causes Israel to run toward it, as it is written "Draw me after You and we will run" — both the drawing and the running being present, first the heaviness requiring drawing and then the willing running.
Shem MiShmuel, Rosh Hashanah explains that the shofar produces an inner voice issuing from the breath of the heart before it is divided through the five articulations of the mouth, and this inner voice awakens the innermost point that is the beginning of existence; since Israel performs teshuvah from the depths of the heart, the shofar-blast enacts a corresponding deed, and — citing the order of Malkhuyot, Zikhronot, Shofarot — one first crowns God as King and only then seeks mercy through the shofar of freedom.
Sulam on Zohar, Vayikra 311 preserves a tradition from the work of Rav Hamnuna Saba that the prayer and the shofar-sound which the righteous person draws forth from their spirit and soul ascends on high, and when that shofar-voice rises, all the accusers are repelled and cannot stand — with the verse "Praised is the people who know the teruah" understood as those who know, not merely those who sound.
Sefer HaChinukh 405:2 grounds the practice in human psychology: every person must arouse their nature to seek mercy for their sins from the Master of Mercy, who is gracious and compassionate, bearing iniquity for those who return wholeheartedly, and the sound of the shofar — and especially the broken teruah sound — greatly stirs the hearts of all who hear it.
Maamarim Melukatim, ש"פ האזינו, שבת שובה ה'תש"מ teaches that the call "Return O Israel unto the Lord your God" addresses the highest level of repentance, for the name "Israel" denotes an exalted spiritual rank, and the two divine names (Havayah and Elohim) allude to the ultimate unification of God's transcendent and immanent dimensions — the deepest knowledge accessible through teshuvah — while Maamarim Melukatim, ליל ערב ראש-השנה ה'תשמ"ג explains that when Jews blow the shofar below, they bring about a corresponding shofar-blowing in the upper realms, since all our mitzvot below effect supernal counterparts above.
Valle, Inyanei Elul veRosh Hashana p. 36 holds that Rosh Hashanah functions as a day of atonement and purification from sins, on which the Holy One sits on the throne of judgment to separate good from evil with divine precision, such that even the righteous cannot rely on their own holiness but only on the sanctity of this annual day that God established for this separation, and Tehillim 81:5 provides the scriptural foundation for this understanding, declaring that sounding the shofar "is a statute for Israel, a judgment unto the God of Yaakov."
Kitzur Shulchan Arukh Yalkut Yosef, Orach Chaim 589:11 rules that it is proper for the one blowing the shofar, or the Torah scholar of the congregation, to address words of reproof to the community before the blasts in order to arouse them to complete teshuvah, citing the Rambam's words that the shofar calls out: "Awaken, you sleepers" — so that the act of sounding and the call to return are inseparably bound together in practice.