This position may seem elusive if you flail about at random, but it comes apart easily with methodical interrogation. Which Black pieces are loose? The bishop and both rooks. (You might look for ways White’s queen could attack any of them and also the king, and come up empty: the needed squares—e.g., d4, d7—are protected.) Since there is more than one loose piece, try attacking two of them at once. Look for a square from which this could be done safely with the queen. You attack rooks from diagonals and bishops from ranks or files, so that the targets can't strike back. The h4 rook and e6 bishop could be attacked from e1, or—if the king weren’t protecting it―from f6. The queen can’t reach either square in one move, so see if it can get there in two moves, with the first one a check or other threat that controls Black's response. Qc3+ requires the Black king to move to h7 or g8; then Qe1 forks the rook and bishop. Black can't escape the loss of a piece. (If White instead plays the forking move Qf6, Black can use the rook to protect the bishop: Re4.)

You also could have found this by just examining every check. The queen has checks at c3, d4, d7, and g6. Only c3 is plausible; all the others result in the queen being captured without an adequate follow up. So you ask what would be possible after the king is forced to move. You are mindful of (a) any loose pieces, (b) the king’s new position on h7, and (c) the White queen’s new position on c3. Qe1 then is simple enough.