Start in the usual way: examine your checks and their consequences. All the checks White has here involve its queen, which can attack the king by moving to b6, d6, c5, c4, c3, or g7. Four of those moves lose the queen on the spot, but 1. Qc4+ does not. Black can’t capture White’s queen in response, so he would have to either move his king or interpose something. Which will it be? Don’t just think of the queen moving to c4; imagine it there—and see that it also would be attacking Black’s rook, which is unprotected. Black would like to avoid the loss of his rook, and so might play Kd7 to protect it while also moving his king out of harm’s way. Now how would the board look? White’s knight is on a light square; so are Black’s king, queen, rook, and bishop—all of which can be forked from c5! The only apparent difficulty is the Black pawn at d6, which protects the needed square. But an early question about any such case is whether the protecting piece is constrained by a pin, and the d6 pawn is indeed pinned to Black’s king by White’s rook. So 2. Nc5+ wins the queen.
If Black doesn't fall for this by playing Kd7 in the first place, then of course White instead uses his second move to take Black's rook. It's an example of a queen fork, which is a theme we will study in detail soon, but the general point already is familiar: sometimes a knight fork (or any other tactic) does its work without ever being carried out. The threat of it forces your opponent to cough up material to avoid seeing it executed. To put it differently, you don't play a move like Qc4+ here hoping that Black will play Kd7. It's great if he does, but you should always assume he will see the trouble coming and will play the best move available to him—probably Qc6, but in any event not Kd7. We say the sequence works here not because we fantasize that Black will play Kd7, but because you are sure to win at least a rook if he plays anything else. Alas, many of the prettiest forks never end up getting played.
Going back to the main idea of the position, the tricky part is that the fork depends on the pin of the d6 pawn, which is visible only if you clearly are imagining both White’s move (Qc4+) and Black’s response (Kd7). It is a perfect example of the importance of visualizing not only the move you imagine making but also the move that comes after it.