Yours, Etc.: Origins and Uses of 8 Sign-Offs
c.1200, "muteness, state of being silent," from Old French silence "state of being silent; absence of sound," from Latin silentium "a being silent," from silens, present participle of silere "be quiet or still," of unknown origin. Meaning "absence of sound" in English is from late 14c.
1560s, intransitive, "become still or silent;" 1590s, transitive, "make silent," from silence (n.). Related: Silenced; silencing.