You are playing as the white pieces and must checkmate to win. Click a square to select a piece and then click the square you wish to move to. To capture a piece, you must click the destination square twice (the first time will remove the opponent's piece and the second will move your piece).
If the board does not change after you try to move, then you have not made the correct move. If you have made the correct move, a button will appear that you must press to show the response from the black pieces. If you press the button and the button remains, then one of your pieces has been captured and you need to press the button again to move the opponent's piece.
If you make a move and nothing happens then that should mean you have checkmated your opponent!