Answer:
Abbott is telling Costello the players on each base, the man on first base is named "Who" so Costello thinks Abbott is asking him who is on first base. So Costello asks "what?" but then Abbott corrects him that "Watt" is the guy that is on second base. This goes back and forth until Costello says "I don't know!" but Abbott tells him the guy named "I don't know" is on third base.