The foundation of the entire discussion is the pasuk in Kohelet 5:1–2, which commands "keep your mouth from being rash" and grounds the reason in the gap between the human and the divine: "God is in heaven and you are on earth" — meaning that awareness of that distance is what should slow speech down.
The Duties of the Heart (Eighth Treatise on Examining the Soul 4:18) draws directly on that pasuk to teach that curtailing speech is praiseworthy precisely because silence has a better outcome, citing the same verse as proof that fewer words reflect proper standing before God.
The Shenei Luchot HaBerit (Aseret HaDibrot, Tamid, Ner Mitzva 11) makes the connection to reverence explicit, ruling that speech must proceed "with deliberateness, awe, and fear" — and that one should think through the meaning of every word before it leaves the mouth, understanding how far its effects reach.
This inward rehearsal before speaking is itself what Mishlei 15:28 identifies as the mark of the righteous person, whose "heart rehearses the answer" before the mouth opens, in contrast to the wicked whose mouth simply blurts, and the Duties of the Heart (Eighth Treatise on Examining the Soul 5:2) frames that constant self-accounting as the very mechanism by which a person never parts from awe and fear of the Almighty who constantly observes him.