Questions about Chess

I just played a game, I ended up with a pawn and my King. The opponent only had a King. My pawn was on h7. The game ended in a draw. How would I have possibly won this game?

With a K plus h-Pawn vs K, the endgame is drawn if the defending King can reach the h, g or f-files. It is only won if the attacking King can reach the g7 square.

This is a draw:

If the White King had started on d4 instead, it is a win because he can reach g7, excluding the Black King from the corner.

Thanks guys! It makes sense!

Is it legal for white king to hit my horse? If in case white king takes black horse it will automatically result in a check. Does black win or does black have to say check in the next move and move another pawn. Can black queen kill white king.

It is not allowed to move the King into check. So the White King couldn't take the Knight if the Knight was protected by Black's Queen. It isn't a legal move. So the game would need to be backed up to the point where the King tried to take the Knight, and White would have to make a different move (or resign) instead.