The foundational grounding for this teaching lies in the Rambam (Mishneh Torah, Repentance 3:4), who writes that although the sounding of the shofar on Rosh Hashanah is a Scriptural decree, it carries an allusion — as if to say: 'Awake, you sleepers, from your slumber, and you who are in deep sleep, rouse yourselves, and examine your deeds and return in teshuvah and remember your Creator'; those who forget the truth in the vanities of time are called upon to look into their souls, improve their ways and deeds, and let each one abandon his evil path.
Likutei Halakhot, Orach Chaim, Laws of the New Year 1:1:4 takes up this tradition directly, teaching that the voice of the shofar alludes to teshuvah — the aspect of 'shapru maaseichem, improve your deeds' — as expressed in the call 'awake, you sleepers, from your slumber,' and that the shofar is therefore the aspect of the World to Come (alma d'atei), which is itself the aspect of Binah, which is the aspect of teshuvah, as is well known.
Likutei Halakhot, Orach Chaim, Laws of the New Year 3:2:1 extends this by explaining that the entire world is aroused to teshuvah at the moment of the shofar blast, since that arousal is the very essence and purpose of the shofar; through it, additional understanding (havanah yeterah) is drawn down, and through that heightened understanding shame becomes revealed on a person's face — the awe and shame that come from the increased insight granted through teshuvah — since the shofar is the aspect of Binah, the World to Come.
Likutei Halakhot, Orach Chaim, Laws of the New Year 4:5:2 deepens the picture further by identifying the shevarim-teruah specifically as the aspect of shame and the broken heart — the sighing and wailing that arise from shame over the greatness of one's transgressions — and holds that this shame is the essential quality of the shofar's voice, which encompasses many dimensions, and for that reason the very word shofar is an expression of splendor and the totality of all these dimensions together.