The key thing for White to notice is that Black’s king and queen are together on the same diagonal with White’s queen bearing down on them from b2. There isn’t yet a pin because White’s knight is in the way, but the potential is there. At first it’s not clear what White could do with the pin, but he is familiar with the concept of the previous positions: if his queen pinned Black’s queen and Black’s queen were protected just by his king, then a check of the king—perhaps by a rook, as in the previous position—would leave the queen loose, and easy prey to a capture.
With that general idea hatched, the rest is a matter of identifying obstacles and eliminating them. The problems are that (a) there is a Black pawn on c4 blocking the c2 rook’s path to the seventh rank; (b) if the rook were able to reach c7, there still would be a Black pawn on f7 that would block the rook’s path to the king; and (c) White’s own knight blocks the pin, as we have seen. The first problem can be dealt with simply: the easiest way to clear an enemy pawn is by taking something it protects, so White plays RxN, Black replies c4xR, and our first obstacle is gone. Now what of the Black pawn on f7 and the White knight on d4? Pairs of problems like this should cause you to ask whether they can be eliminated together: move the knight to a square the bothersome pawn protects, and perhaps they both will end up out of the way. Thus White plays Ne6+, which forks Black’s rook and king and also unmasks the pin of Black’s queen by White’s. If Black moves his king, his queen is lost to QxQ. So Black instead plays f7xN. Now all three of the obstacles listed earlier have been eradicated. White can play Rc7+, forcing Black to move his king and leave his queen to be taken with QxQ.
This position is a good study in the value of seeing an idea suggested by a pattern and then calmly removing the impediments to it one by one. The pattern is the potential pin of a queen adjacent to its king. The idea is to push the king away from the queen so the queen can be taken for free. The impediments are the three obstructions on the needed lines. As we see, they can be eliminated with forcing moves that give Black no time for counterplay. Black need not take the bait, of course; he does not have to recapture after White's initial RxN. White gains in any event.