stalemate requirements

It was your opponent's move and there were no legal moves available to him. You just have to be careful not to leave him without a move, otherwise he can't play, you can't play because it's not your go, and so the game is drawn.

Stalemate occurs when the person whose turn it is can't make a legal move and isn't in check.  In the game you mentioned, your opponent only had a king, but the king couldn't move anywhere without putting himself in check.

He couldn't move to g4 or h4 because of the rook on a4.  He couldn't take the pawn on g5 because it was protected by the knight.  He couldn't move to g6 because of your bishop and he couldn't move to h6 because of your knight and pawn.  Since no piece was currently checking the king, the result was a stalemate.

A tricky opponent will sometimes try to trap you into stalemating him as a last-ditch attempt to salvage a draw from his imminent loss.

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