1830, American English, said to be a variant of Kentish dialect feeze "to frighten, alarm, discomfit" (mid-15c.), from Old English fesian, fysian "drive away, send forth, put to flight," from Proto-Germanic *fausjan (source also of Swedish fösa "drive away," Norwegian föysa). Related: Fazed; fazing. Bartlett (1848) has it as to be in a feeze "in a state of excitement." There also is a nautical verb feaze "to unravel" (a rope), from 1560s.
"to synchronize, adjust the phase of so as to synchronize," 1895, from phase (n.) in the physics sense of "particular stage or point in a recurring sequence of movement or changes" (1861). Earlier as a bad spelling of faze. Meaning "to carry out gradually" is from 1949, hence phase in "introduce gradually" (1954), phase out "take out gradually in planned stages" (1954). Related: Phased; phasing.