user17929: 
The pawn move from h2 to h3. I do not understand the reason behind this.

user825: 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.

17929: <p><img src="https://i.stack.imgur.com/AhYE9.png" alt="enter image description here"></p>

<p>The pawn move from <code>h2</code> to <code>h3</code>. I do not understand the reason behind this.</p>
 825: <p>It's a <strong>common</strong> idea in the position. It's there to prevent <code>Be3 Ng4</code>. 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 <code>Be3-Qd2-Bh6</code> or <code>Be3-Bd4-Bxg7</code>.</p>

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

<pre><code>9.Be3 Ng4! 10.Qe2 Nxe3!
</code></pre>

<p>Black's position is fine with the strong g7 
bishop unchallenged. <code>9.h3</code> is a good move.</p>
