You don't need to consider every state individually to solve a game. For example if you have a game on a 100x100 board and the only goal is to move your piece to the other side you can know you have found the optimal solution if you can do it in 99 moves (assuming that you move 1 per turn).

In the chess scenario if you can prove that there is a set of moves where you win no matter what the other colour does then you have solved the game. You don't need to consider every possible game, just the games reachable within this set of moves.

Solving chess would be proving that "From starting position black can win every time" (or the same for white). You don't need to prove how to win from every possible board state.