"long, loose outer garment reaching almost to the floor, worn by men or women over other dress," late 13c., from Old French robe "long, loose outer garment" (12c.), from a Germanic source (compare Old High German rouba "vestments"), from West Germanic *raubo "booty" (cognate with Old High German roub "robbery, breakage"), which also yielded rob (v.).
Presumably the notion is of fine garments taken from an enemy as spoil, and the Old French word had a secondary sense of "plunder, booty," while Germanic cognates had both senses; as in Old English reaf "plunder, booty, spoil; garment, armor, vestment."
The meaning "dressing gown" is from 1854; such extended senses often appear first in French, e.g. robe de chambre "dressing gown," robe de nuit "nightgown." From c. 1300 in reference to official vestments and thus indicative of position or membership in a religious order, guild, etc.; metonymic sense of The Robe for "the legal profession" is attested from 1640s.
"to clothe," especially magnificently and ceremonially, c. 1300 (implied in robed), from robe (n.). Related: Robing; robery.