Losing the king constitutes loss of the game. Checkmate is quite simply a position where that loss is unavoidable. In the given position if your opponent moves Kg6 then you capture his king with your king BEFORE he can recapture with the knight. In effect he loses before you and has therefore lost the game.
Clarification on a rule

seems like he wants to be able to move his king into check, but not allow you to do likewise. The simple fact is, the first person to lose their king, loses. He could play this in quickplay I think, and you would just capture the king. Game over. He can move the pieces around all he likes afterwards; the game has already finished.
It's the same as when you must answer a check. You cant respond to a check by checking or "checkmating" your opponent, because your king is still in check and will be the first to fall. In a quickplay, the person giving the "checkmate" would lose, even though their opponent only gave a simple check.
The simple answer is:
Why cant I go there?
You cant go there because I will take you.
You cant take me because then I will take you.
You cant take me because I have won, the game is over.
Sorry if there are weird spaces in the last bit, I have no idea why it has left gaps. I didnt enter any spaces.
Think of your king as the "power supply" that keeps all your other pieces going. If he moves his king beside yours, you could simply take his king. At that point all his pieces turn into dust and vanish from the board - so he has no knight left to recapture your king with!
A simpler explanation is: the rules say you cannot EVER place 2 kings on adjacent squares, period.

LOL. I've had the same question put to me and it doesn't matter how you answer it, such people cling to the notion. You might be best to play cards with him.
It's analagous to a guy I used to play who insisted that on your first move, and only one your first move, you could move the a- and h- pawns simultaneously one square. When I couldn't talk him out of it I responded by moving my b- and g- pawns one square for a pair of cheap fianchettos. He got the message and stopped doing it.
A friend of mine and I were engaged in a game of chess and he has asked a question that I'm not sure exactly how to answer. The situation is essentially as follows:
In this position, he wants to know why he can't play Kg6, since my king could not take his king because it is "protected" by the knight on e7. I have explained that it is illegal to move his king into a position where his king can be captured by an opposing piece, but he says my king would not be able to capture his king because I would be moving my king into check. So essentially he is saying my king should not be able to capture his king if his king is protected by another piece... or to put it a different way, it shouldn't be considered putting his king into check if it would be illegal for me to capture his king on that square.
We've circled the issue a few times on this one, and I'm wondering if someone can explain the situation in a new way or a more clear way to clarify why Kg6 is an illegal move.
Thanks.. ogin