'unmuzzle' is formed by the prefix 'un-' (meaning 'reverse' or 'remove') combined with 'muzzle' (from Middle English 'mussel'/'muzel'), where the element 'un-' meant 'to reverse or remove'.

'muzzle' comes from Middle English (mussel, muzel) and Old French 'musel' meaning 'snout guard'; 'un-' is an Old English prefix. The modern compound 'unmuzzle' is a straightforward negational/ reversive formation in Modern English from these elements.

Initially compounds of 'un-' + a noun like 'muzzle' had the literal sense 'remove the muzzle'; over time the verb 'unmuzzle' acquired an extended, figurative meaning 'remove restrictions on speech' that is widely used in modern contexts.