Old English scrud "a garment, clothing, dress," from West Germanic *skruthan, from Proto-Germanic *skrud- "cut" (cf. Old Norse skruð "shrouds of a ship, tackle, gear; furniture of a church," Danish, Swedish skrud "dress, attire"), from PIE *skreu- "to cut" (see shred (n.)).
Specific meaning "winding-sheet, cloth or sheet for burial," to which the word now is restricted, first attested 1560s. Sense of "strong rope supporting the mast of a ship" (mid-15c.) is from the notion of "clothing" a spar or mast; one without rigging was said to be naked.