From Middle French bider (“to trot”), of unknown ultimate origin, + -et. Possibly related to Medieval Italian bidetto (“small horse”), itself probably related to Proto-West Germanic *biti (“bite; horse's bit”); or, possibly from a lost Middle French rabider (“go quickly, violently”), a descendant of Latin rabidus (“furious, fierce”), with loss of the initial prefix.
Modern sense derives from analogy with the straddling of a bidet and the straddling of a small horse.