late 14c., mosel, "device put over an animal's mouth to stop it from biting, eating, or rooting," from Old French musel "muzzle," also "snout, nose" (12c., Modern French museau), from muse "muzzle," from Gallo-Roman *musa "snout" (source also of Provençal mus, Old Spanish mus, Italian muso), a word of unknown origin, possibly related to Latin morsus "bite" (but OED finds "serious difficulties" with this).
Meaning "projecting jaws and nose of the head of an animal" is from early 15c.; sense of "open end of a firearm" is recorded from 1560s. Muzzle-loader "gun loaded from the muzzle" (opposed to breech-loader) is by 1858.
"to put a muzzle on, bind or confine the mouth of to prevent from biting or eating," early 15c., moselen, from muzzle (n.). Figurative use, "to gag, silence," is from 1610s. Related: Muzzled; muzzling.