late 14c., disporten, "to divert (from sadness or ennui), cheer, amuse gaily," from Anglo-French disporter "divert, amuse," Old French desporter "to seek amusement," literally "carry away" (the mind from serious matters), from des- "away" (see dis-) + porter "to carry," from Latin portare "to carry" (from PIE root *per- (2) "to lead, pass over").
Compare disporter "a minstrel or jester" (early 15c.), also Latin deportare "to carry away, transport," in Medieval Latin also "divert, amuse." For a similar sense evolution, compare distract, divert, transport (v.). Intransitive sense of "to play, sport" is from late 14c.
c. 1300, "activity that offers amusement, pleasure, or recreation," from Anglo-French disport, Old French desport, from disporter/desporter "divert, amuse" (see disport (v.)). From late 14c. as "a sport or game; the game of love, flirtation."