Confused about a checkmate that involves a pin and protection

Newbie question here: I found the checkmate below after several failed moves while playing puzzles. If the Rook on g2 is not able to move because it is pinned by the Queen on h3, then how can the Rook "protect" its Queen on h7? If it is illegal for this Rook to move out of the pin and cause a check on its own King, it does not seem possible that the Rook is truly protecting Queen on g7 if the Rook cannot move.

Your thoughts?

Here's the easy way to think about this: if the Black king took the rook, which king would be captured first? Sure, White's rook is "pinned", but it would be able to take the Black king before Black could take the White king.

Therefore, the queen on g7 cannot be captured. Checkmate.

1.4 The objective of each player is to place the opponent’s king ‘under attack’ in such a way that the opponent has no legal move.

1.4.1 The player who achieves this goal is said to have ‘checkmated’ the opponent’s king and to have won the game. Leaving one’s own king under attack, exposing one’s own king to attack and also ’capturing’ the opponent’s king is not allowed.

https://handbook.fide.com/chapter/E012023

All you need to understand is that pinned pieces can still give a check. It is illegal to move your king into check, so you can't take this piece with the king. 

I think it helps to remember that chess is a turn-based game: I move, the you move, then I move, etc. Keeping that in mind, it's not hard to see that in #1. if 1...Kxg7 2.Rxg7, the game is over Black doesn't get another turn

the check is immediate. the rules state that it must be dealt with immediately to a satisfactory conclusion. with check by the queen, the black king has no legal move. it cannot take the queen because it will be moving in to check by the rook. that fact that rook is pinned does not matter.

Thanks everyone... your comments cleared this up for me.