[band of silk, strip of cloth], 1550s, "a band worn across the body or over the shoulders," probably from Old North French escarpe "sash, sling," which probably is identical with Old French escherpe "pilgrim's purse suspended from the neck," perhaps from Frankish *skirpja or some other Germanic source (compare Old Norse skreppa "small bag, wallet, satchel"), or from Medieval Latin scirpa "little bag woven of rushes," from Latin scirpus "rush, bulrush," which is of unknown origin [Klein].
It also is attested in early Modern English as scarp. OED points to "the change of the initial p into f after liquids". As a cold-weather covering of warm and soft material for the neck, by 1844. Plural scarfs began to yield to scarves early 18c., on model of half/halves, etc.