I'm 90% sure this refers to a wrestling move. In the sport of wrestling, you win by holding the opponent's shoulders down on the mat for for a short time, perhaps two seconds. This is called "pinning" the opponent. It sounds as if they are describing an old-fashioned circus sideshow or carnival, and that one of the features was a strongman--and he would let you wrestle with him, if you would agree to pay him a dollar if he could pin you. I never heard of this but it sounds like the kind of thing they did at carnivals.