"to strip off" the skin, bark, or rind from, developed from Old English pilian "to peel, skin, decorticate, strip the skin or ring," and Old French pillier, both from Latin pilare "to strip of hair," from pilus "hair" (see pile (n.3)). Probably also influenced by Latin pellis "skin, hide." Related: Peeled; peeling. Intransitive sense of "to lose the skin or rind" is from 1630s.
The figurative expression keep (one's) eyes peeled "be observant, be on the alert" is from 1853, American English. Peel out "speed away from a place in a car, on a motorcycle, etc.," is hot-rodders' slang, attested by 1952, perhaps from the notion of leaving behind a "peel" of rubber from the tire as it skids. Aircraft pilot phrase peel off "veer away from formation" is from World War II; earlier American English had slang peel it "run away at full speed" (1860).