This is a really interesting question.
The Ramban on Parshat Emor asks the same exact question. The Torah is very vague about what exactly goes on on this day, and only says "it should be a day of Teruah," not even defining what a teruah is. It also says "a remembrance of Teruah" which doesn't seem to help us more.
The Ramban says:
לא פירש הכתוב טעם המצווה הזאת למה התרועה, ולמה נצטרך זיכרון לפני השם
ביום הזה יותר משאר הימים, ולמה יצווה להיותו מקרא קדש כלל. אבל מפני
שהוא בחדשו של יום הכיפורים בראש החודש נראה שבו יהיה דין לפניו יתברך כי
בם ידין עמים, בראש השנה ישב לכסא שופט צדק, ואחרי כן בעשרת הימים ישא
לפשע עבדיו. נרמז בכתוב העניין כאשר נודע בישראל מפי הנביאים ואבות
קדושים.
A summary in English: The Torah never tells us what the point of the day is, why we have a Teruah, or why we need a zikaron, and furthermore, the Torah never even gives any reason at all for the holiday. So we have to see that Yom Kippur happens to be in the same month, and we figure out that Rosh Hashana must be linked to it. And from this hint, all of our ancestors and the Neviim understood that Rosh Hashana is the day where Hashem starts to judge us.
In his explanation, he also explains that the remembrance is to remember us for the good and judge us fairly (obviously leaning towards Middat Harachamim for Klal Yisrael, like the Yerushalmi).
Additionally, in Parshat Noach, he gives another reason why it's the Rosh Hashana: it was the day the world was created.
