ChessPuzzle uses a built-in engine called "Sjeng".   Sjeng is a powerful chess program developed by Gian-Carlo Pascutto that plays expert level chess even at a setting of only 2 or 3 seconds per move.

The engine can play out your solutions with you.   After you make your move, the engine will do its best to refute it.   It is also handy at suggesting moves.   To get a suggestion, click the light bulb icon.   A translucent red arrow will light the way.

If you use the light bulb icon for the first move, the suggestion comes from the "best move" solution that came with the puzzle instead of the engine.   Multiple best moves are possible and all of them will be displayed. Using the light bulb will not get you the big checkmark.

Technically, once you make the correct first move, you're done and can just go on to the next puzzle. But if you can smell a mate, go for it!