It doesn't stand for anything. It is just a common thing in the U.S. (and possibly elsewhere) to use different numbers of A's to classify things, especially in sports.
One reason this is common is because in professional baseball, the minor leagues are classified (from weakest to strongest) into class A leagues, class AA leagues, and class AAA leagues. They used to go from class D up to class A, but as new stronger leagues formed, they had to create a class above A, so they went to AA. And then they later created another class above that called AAA. And eventually, all the classifications below A were dropped or renamed, so now everything is just based on the number of A's. This has also been picked up by high school sports, where schools are classified from 1A up to 4A or 5A based on how big the school is so that they can compete against similarly sized schools (the exact classifications can differ by state, but most states use the 1A, 2A, etc classifications).
So the "A" is superfluous. It's just a tradition to classify things that way. With regard to yoyoing specifically, it goes back to when contests were only A (one-yoyo tricks) and AA (two-yoyo tricks). AA was (at that time) harder and respected as the higher level of competition, so AA was originally a higher classification than A. The names also reflected that single-A was one yoyo and double-A was two yoyos. When two-handed string tricks, offstring, and counterweight play were added as competitive divisions, they just extended the A naming scheme to 3A, 4A, and 5A.