The pasuk in Vayikra 23:24 designates the first of the seventh month as a day of "זִכְרוֹן תְּרוּעָה" — a remembrance of the teruah blast — without explaining why this particular day requires such remembrance before God.
The Ramban (Vayikra 23:24) explains that "זִכְרוֹן תְּרוּעָה" means that we sound the teruah and it becomes a remembrance before God, parallel to the trumpets sounded in the Mishkan that similarly served as a remembrance before God (Bamidbar 10:10); he notes that the Torah does not spell out the reason for the teruah on this day specifically, nor why a remembrance before God is needed on this day more than any other — but from the fact that it falls in the month of Yom Kippur on the first of the month, it appears that this is a day of divine judgment.
רמב"ן, דרשה לראש השנה, וראה שם עוד וכן בתורת האדם שער הגמול develops this understanding explicitly, stating that "זִכְרוֹן" means it is a day of din — judgment — and that this is precisely what the Sages encoded in the liturgical phrase "את יום הזכרון הזה," meaning "this day of judgment"; he goes on to treat the teruah's instrument separately.
The Gemara in Rosh Hashanah 16a frames the Remembrances (זִכְרוֹנוֹת) as one of three sets of verses recited on Rosh Hashanah so that one's remembrance will rise before God for good, with the shofar serving as the vehicle for that remembrance — and connects the ram's horn specifically to the Akeidah, so that God will recall the merit of Yitzchak's binding.
Rashi (Vayikra 23:24) similarly anchors the זִכְרוֹן to the verses of Zichronot and Shofarot recited in the Musaf prayer, and to the memory of the Akeidah through the ram's horn, while אברבנאל, ויקרא כג כד elaborates that Rosh Hashanah involves two judgments — a universal one governed by natural order and a particular one for Israel — and that the shofar's combination of teruah and tekiah corresponds to both, with the opened books applying to Israel who can rise above celestial decree through their attachment to God.
רש"ר הירש, במעגלי שנה א עמוד סט, וראה שם עוד ובעמוד פה broadens the meaning of Yom Hazikaron to encompass a renewal of awareness of God's kingship, His providence, humanity's frailty and dependence, and the covenantal relationship — making the day a comprehensive re-orientation of memory toward all that is essential, from God's greatness to human accountability.