A piece protected by its king is a vulnerability: the king can only provide protection so long as it is right next to the piece it guards; and in response to an attack the king often has to move since adding defenders to it is no help. So when you see a piece so protected, it's especially useful to look for any checks you can give; whatever else they do, the checks may leave loose whatever the king meant to guard.
Turn to the position on the left. Question 1: What does Black attack? Answer: White’s rook; nothing more. Question 2: What guards White’s rook, preventing RxR? Answer: White’s king. Question 3: Can White’s king be checked and thus driven away from the rook, leaving it loose? Answer: Yes, with Bd3+. The king moves to g1, and now Black wins the rook with RxR+.
On second thought, however, don’t jump to the conclusion that a check will force the king to move; consider whether your opponent has other options. Here White also would have the option of Re2, blocking the check with his rook. But now the rook would be attacked twice and still defended just once, so Black again wins it with RxR. Black's initial Bd3+ thus works fine, but we'll want to stay in the habit of looking at all of an enemy's possible replies to threats like this.