Black might productively begin by inspecting checks like Qf1 or Qe1. Either move forces White to play Kb2 or (if Black starts with Qf1) Kd2. But then what should Black do? You will see nothing more if you confine your attention to follow-ups with your queen. The key to the position is to notice the kernel of a discovery that exists from the beginning (and which is kept intact by Black's Qf1)—the two queens with a Black pawn between them. Or you might see the solution by simply asking what check Black can give after White has moved his king and remembering to include pawn moves in your thinking. The point either way is that if Black starts with 1. …Qf1, he can follow it with the pawn discovery c4-c3+, permitting him to play QxQ next move.
1. …Qe1, by the way, not only doesn’t work but results in mate for White. Notice that White is close to mating already with his rook and queen; he does so as soon as Black runs out of checks that hold the initiative. Thus 1. ...Qe1; 2. Kb2, c4-c3+; 3. Kb3, and Black has no way to stop White from playing Ra7+. If Black replies by moving his king to d8, White mates with Qd7 or Qb8; if Black instead moves his king to c8, White mates with Qe8. Studying the operation of White’s queen and rook in this position is highly worthwhile, as their ability to mate by trapping the king in this way (sealing off two ranks) comes in handy. What also makes it work, of course, is the Black king’s inability to escape toward the center with Kd6.