It has nothing to do with getting the girl.
In rodeos, the pickup men are the riders - normally two - who are positioned in the arena during a cowboy's ride. Their task is to help the rider dismount, release the flank strap and then get the bronc or bull out of the arena quickly and safely. The job requires athleticism and very good horsemanship. (Some larger rodeos may have more than two pickup men because of the size of the arena.
They're vital for the safety of not just the cowboys but the broncs, but pride themselves on being almost invisible.