Why is this a mate?


if opponents queen mated then your king could not capture the queen because opponents “pinned” rook still controlled the capture square- and a king cannot place itself in check (even if checking piece is pinned)
A pinned piece can check or checkmate in a variety of ways. A pinned piece could restrict where your king can move to as part of a checkmate. You could even move a piece into a pin and check your opponent’s king while you do it.
I've always thought of it dropping the terms check and checkmate completely, and thinking of it as whoever first captures the opponent's king wins. This doesn't have any practical effect on the game other than the fact that it is extended by 1 move.
For example:
In this case, White has just moved their queen and "checkmated" Black. However Black moves, White will be able to capture their king and win.
In this case, if Black's king captures White's queen, the rook would capture Black's king. Black would then capture White's king, but it doesn't matter since their king had been captured first.