c. 1400, from Old French fausset (14c.) "breach, spigot, stopper, peg (of a barrel)," which is of unknown origin; perhaps diminutive of Latin faux, fauces "upper part of the throat, pharynx, gullet." Not in Watkins, but Barnhart, Gamillscheg, and others suggest the Old French word is from fausser "to damage, break into," from Late Latin falsare (see false).
Spigot and faucet was the name of an old type of tap for a barrel or cask, consisting of a hollow, tapering tube, which was driven at the narrow end into a barrel, and a screw into the tube which regulated the flow of the liquid. Properly, it seems, the spigot was the tube, the faucet the screw, but the senses have merged or reversed over time. OED reports that faucet is now the common word in American English for the whole apparatus.