smooth board, chess-board, chess
枰 centers on a flat wooden surface used for board games: a smooth board, especially a chessboard, and by extension the game of chess itself.
枰 combines 木 (wood) with 平 (flat, level), suggesting a flat wooden board. The exact historical development is uncertain, but the component choice fits the meaning of a smooth game board.
A wooden 木 board that is flat and level 平: picture a smooth wooden chessboard ready for a game.
For ヘイ, imagine a smooth wooden board painted in shades of hay: hay -> ヘイ, and the flat board is ready for chess.