What does Boxing Day have to do with boxing?
1530s, "a running (usually at full speed), a course" (especially of the sun, etc., across the sky), from Middle French carriere "road, racecourse" (16c.), from Old Provençal or Italian carriera, from Vulgar Latin *(via) cararia "carriage (road), track for wheeled vehicles," from Latin carrus "chariot" (see car). Sense of "course of a working life" first attested 1803.
1590s, "to charge at a tournament," from career (n.). The meaning "move rapidly, run at full speed" (1640s) is from the image of a horse "passing a career" on the jousting field, etc. Related: Careered; careering.