Why is this a checkmate?

Black has the next turn here. I thought kings couldn't attack each other. Couldn't black king take the white queen and then the black king would be safe?

The queen is defended by a king, black king can't touch it, it is an illegal move.

It is true that king can't put the other king in check, but that is because it is illegal to put your king on a square next to the other king, as the other king controls that square. So you can't put black's king on g1 while white's king is on f1.

There is a logical reason (beyond the illegality) why kings cannot attack each other.

In any case, capturing the queen brings the king to each other, so it is illegal, just like nklristic mentioned.

But....how do you explain in chinese chess,

emperor can checkmate opposing emperor.

This one is going to be hard to explain in simple terms. I cannot think of an analogy for that at the moment.

It's awkward how the emperor cannot move out of his comfort zone, yet can "fly over" to defeat the opposing emperor.

the kings cant stand together, otherwise they are both attacked

I guess one way to think of it is that each king has an anti-king force field around it that only affects opposing kings and goes to the mid-point of every square next to the king.  An opposing king would have to center itself in a square it is moving to and thus cannot move next to the king.

If the king moves near the opposing king, that would be moving into check and is therefore not allowed.

That simple explanation was not understood by the OP and he brought up a Chinese chess tangent.

I'm not all too familiar with Chinese chess, so correct me if I'm wrong, but if a move is made and the two emperors are facing each other the emperor can capture the opposing emperor. It seems to me that creating this situation in the first place would mean moving your king onto an attacked square, so in practice it wouldn't be allowed even if not strictly prohibited. This is essentially the same as in chess.

The emperor cannot move as to face the opposing emperor, for it would be captured on the next turn.

As such, the king cannot move near the opposing king as it would be captured on the next turn.

I'm not sure why Chinese chess is brought into this.

For standard chess the rules are easiest to understand if you just ignore check altogether.  The game is lost by the person that loses his King first.  In chess the has been made to end the game 1 move (1 ply to be specific) before the King is taken, but the result will be similar to a game without the rule of check.  In OP's example, the Black King has 3 moves, two of which would result in being taken by the Queen, the third one (taking the Queen), would result in being taken by the King. 

The only exception is s stalemate, where all moves result in being taken, therefore all moves are illegal thus ending in a draw.

This is reguler chess not chinese chess!

It is illegal to move into check.

It's chinese chess. Its different from actual chess

What Morphyblanca said. It is illegal for the king to move into **any** check. When one king moves next to another king, the other king would be able to take it, thus it's an illegal move. It doesn't matter that the king would have to move into check to take it because the initial move is always illegal. Always.

To the players who play Chinese chess, it is the "actual chess".

Kings like to be next to queens, they don't like hanging around with each other though for some reason 

wrong varient guys

The way I do it is just imagine that the kings could attack each other and so that is why they can't be next to each other. Not strictly accurate but it works for me.