It all starts with seeing a loose piece. Here Black has left his rook loose, so it's a target and the focus of operations. White can attack it with his queen but can’t give check at the same time, and the check that is available to the queen—Qe8—won’t change that. Yet the fork seems so close: Qd3 attacks the rook and would attack the king if only it were on h7. Look for checks by other White pieces that might push the king around, and you find just one: Rb8+, which forces the reply Kh7. Qd3+ follows, of course, winning the rook.