late 14c., "carried away in an ecstatic trance," from Latin raptus, past participle of rapere "seize, carry off" (see rape (v.)). A figurative sense, the notion is of being "carried up into Heaven" (bodily or in a dream), as in a saint's vision.
The Latin literal sense of "carried away" also was in English from 1550s. Essentially an alternative past participle of rape, in 15c.-17c. the word also sometimes could mean "raped." The sense of "engrossed" is recorded from c. 1500.
As a Latin past-participle adjective, in English it spawned unthinking the back-formed verb rap "to affect with rapture," which was common c. 1600-1750. Before that, there was a verb rapt "seize or grasp, seize and carry off; ravish" (1570s), also "enrapture, transport as with ecstasy" (1590s). There also was a noun rapt in 15c. meaning both "rapture" and "rape."