A paradox is an observable fact or logic that seems to defy explanation. In the case of archery, back when the term was coined, bows were made of wood and most did not have arrow shelfs cut into the bow. Thus, when the arrow was placed on the string/bow, it pointed to the left for a right hand shooter. Common sense would suggest that the arrow would careen off to the left when shot, yet observation showed that the arrow flew true to the target at which it was aimed.
That's the archer's paradox. The apparent unexplainable arrow flight. Why did the arrow go straight, which observation proved to be the case, when common sense would suggest it should shoot to the left. Of course, through high speed photography, and just general sleuthing, we now know the answer to that paradox. The arrow strikes the target because, when properly spined, it bends/flexes just the right amount around the riser when shot.
It's the bending of the arrow around the riser that explains the paradox. The arrow does not "paradox" around the riser, as I've seen some state in the past.