The sefer LeDofkei BaTeshuva by Rabbi Uri Teegar on Rambam's Hilchos Teshuva 1:1, in Biurim s.v. על לשון כל ישראל (near § 65, it's too far to see for free on Otzar HaChochma) brings the Sefer HaChinuch § 364, who explains the reasons for vidui (verbal confession): (1) expressing one's sins reveals one's thoughts and beliefs, such that it's clear they don't think their sin went unnoticed by Hashem. As well, (2) through expressing it they'll regret their actions more and will refrain from transgressing again. All of this can be accomplished through a shaliach's confession. Although a person can have regret even without verbalizing it, explicit vidui will make it have a stronger impression.