The title is slightly misleading. It's not storing one file in one chess game, but storing one file in a sequence of chess games.
The technique has not much to do with chess but applies to any game, as long as you have a deterministic move generator.<p>If in the current game state that produces a list of 2^k <= n < 2^{k+1} moves, then you get to encode the next k bits of the file, and change the game state to that after the corresponding move. Whenever the game ends, you start a new one.