The Shechinah speaking through Moshe's voice

איתא במד' כשרצה הקב"ה לומר אנכי צוה למרע"ה לירד שלא יאמרו בני ישראל שהוא דברי משה ע"ש.

It is brought in the Midrash: when Hashem wished to say "Anochi" (I am Hashem your God), He commanded Moshe Rabbeinu to descend, so that Bnei Yisrael would not say that these were the words of Moshe; see there.

At the very moment of giving the first Dibber, Hashem sent Moshe down among the people, so they would clearly understand that the voice speaking was Hashem's and not Moshe's own.

מוכח מזה כי שכינה מדברת מתוך גרונו של משה דאל"כ הרי יבחנו בטביעות עינא דקלא.

This proves that the Shechinah speaks from within the throat of Moshe — for otherwise they would distinguish [it] by the recognizable identity of the voice.

The Sefas Emes reasons: if the heavenly voice had sounded different from Moshe's, no confusion could arise, and Moshe's descent would be unnecessary. The very concern shows that the Shechinah ordinarily spoke through Moshe's own throat, in his own recognizable voice.

רק שבאמת הי' זה הקול כקולו של משה.

Rather, in truth this voice was like the voice of Moshe.

Indeed the Divine speech at Sinai issued in a voice indistinguishable from Moshe's own — which is exactly why the people might have mistaken it for his words.

ולכן כתיב וגם בך יאמינו לעולם כי מאלה הדיברות ראו כי נבואת מרע"ה הוא מפי הקב"ה.

And therefore it is written, "and they will also believe in you forever" (Shemos 19:9) — for from these Dibros they saw that the prophecy of Moshe Rabbeinu is from the mouth of Hashem.

By experiencing at Sinai that Hashem Himself spoke through Moshe's voice, Bnei Yisrael gained permanent emunah that all of Moshe's later nevuah likewise came directly from Hashem.

וכ"כ בעבור ישמע העם בדברי עמך:

And so it is written, "so that the people may hear when I speak with you." (Shemos 19:9)

The purpose of the revelation was precisely that the people hear Hashem speaking "with you" — through Moshe — establishing forever the authenticity of his prophecy.

Summary: Hashem sent Moshe down at the moment of "Anochi" so the people would not mistake the words for Moshe's own — proof that the Shechinah ordinarily spoke through Moshe's very throat, in a voice like his. By witnessing this at Sinai, Bnei Yisrael came to believe forever that Moshe's prophecy issues directly from the mouth of Hashem.