What is the idea behind h2-h3? The pawn move from h2 to h3. I do not understand the reason behind this.
It's a common idea in the position. It's there to prevent Be3 Ng4. Allowing Ng4 with the bishop on e3 is not good; White's dark bishop should be exchanged with Black's strong g7-bishop, something like Be3-Qd2-Bh6 or Be3-Bd4-Bxg7.

Please note White played 10.Be3 immediately next. Without 9.h3, something like this would have happened:

9.Be3 Ng4! 10.Qe2 Nxe3!


Black's position is fine with the strong g7 
bishop unchallenged. 9.h3 is a good move.