You see Black’s king and queen adjacent on the same diagonal and so immediately think about pinning possibilities. The only piece White has available for the purpose is his queen, which he can play to a3. Obviously Black can just reply QxQ; the question is what would then become possible for White. Imagine the board as it would look after that little sequence and you see three White pieces trained on the Black king’s general position: the bishop on b5, the rook on d1, and the knight on e4. Since White by assumption would be losing the queen, the question had better be whether he can use those pieces to mate afterwards. Think in checks. White would have just one: Rd8+. To this Black would have to respond Ke7. Now notice that Black’s king would be trapped on the seventh rank: White’s rook would seal off the eighth rank, and his knight would seal off the king’s only flight squares on the sixth. So now what? White plays the simple Re8#, a variation on the standard mating pattern where the rook mates with help from the bishop, which protects it and seals off critical flight squares.
The point of all this is that White can attack Black’s queen without fear that Black will play QxQ, for Black's queen is frozen in place by the mate threat. Black will have to reply to the pin by playing g7-g6 to create an escape route for his king. White’s task then is to go to work destroying the pinned queen’s defenders—the king and the knight on g8—so that the pin is made profitable. First White uses Rd8+, forcing Black’s king away to g7. Then he plays RxN+, which loses the rook for a knight (after Black plays RxR) but destroys the last of the Black queen’s defenders. Notice that although the pin of Black’s queen was dissolved back when the king moved away from f8, since White has operated with checks Black has had no time to play QxQ. Now the Black queen has been left loose, so White takes it with his.
Once you realize how much work is being done here by the threat White’s rook creates on the d file, a different idea may occur to you: Black can disrupt White’s plans by replying to Qa3 with Bd7, blocking the White rook’s path. It looks good because White can’t afford to play RxB; that would permit Black to play QxQ without worrying about the mate threat discussed a moment ago. (Do you see why? If White then tries Rd8+, he loses the rook to RxR; when Black moved his bishop to d7, he freed his rook on a8 to protect the back rank.) But White has a different winning sequence instead. He plays QxQ, allowing the recapture NxQ—and then he has BxBd7, as Black’s bishop is left loose. If Black recaptures the queen with his king rather than his knight, White plays RxB+ instead; Black’s king can’t recapture because White's rook has protection from his bishop.