Why can the white king not take the queen here?
Because that would move the king into check - the queen is defended by the bishop on c5.
More generally there is no problem with a king taking a piece that is checking it, provided the capture does not move the king into check or break any of the other laws of chess.
A king can only capture unprotected pieces. In this case, the Black queen is protected by the black-squared bishop.
If a king moves into a square where an enemy piece (the bishop) can recapture him, he is "moving into check," which he can't do. If a player can't move out of check with his king, either by moving away, capturing the checking piece or "interposing" (which doesn't work when the checking piece is adjacent) he loses the game.
But here, the king can move out of check by going to c3. Put the Black knight on b5 or d5, and White is check mated.