Advantage of Diagonal Opposition Suppose there are only two pieces on the board, black king and white king. Black king is at g7 and white king is at e5. It's Black's turn to move. So in this case white has the opposition. Black now plays Kg6.
Can white reach any of the last three squares f8 or g8 or h8. If the answer is "no", then what's the advantage of having diagonal opposition ?
Diagonal opposition can be thought of as a means to an end. No matter where your opponent's King moves, you can get direct opposition sooner or later. Examples from your position:

1...Kf7 2. Kf5 (White gets the direct opposition).

1..Kg6 2. Ke6 (White gets the direct opposition).

1...Kg8 2. Ke6 Kh8 3. Kf6 Kh7 4. Kf7 (White gets the direct opposition).

As per your question regarding f8, g8, and h8, White could not reach the squares even if he had direct opposition in a similar position. Put White's King on e6, and Black's King on g6 (Black to move). You'll find that Black can defend the 3 squares, despite White having the opposition.

The purpose of opposition is that it forces your opponent to move his King out of the way, letting you pass in one way or the other. Note that the way your opponent moves his King out of the way is up to him, so he could protect f8, g8, h8 indefinitely by keeping his King there. However, that would let your King pass his by playing Kf5, Kg4, Kh4 (and reaching the h-file, if that was your goal).