What is the intuitive meaning of "leaf weight" in xgboost
I would read the paper (section 2.2 specifically) for a more concise presentation and formal definition of the terms.

You need to define the loss function $\mathcal{L}$ for your problem. Then $g_i$ and $h_i$ are the first and second derivatives, respectively, of $\mathcal{L}$. The optimal weights are then (equation (5) or slide 27).
$$
w_j^* = -\dfrac{\sum_{i \in I_j}g_i}{\sum_{i \in I_j}h_i + \lambda}
$$
where $\lambda$ is a regularization parameter.

So those weights give you the minimum of (the second order approximation to) $\mathcal{L}$.

Edit:

Since the weights are given in the example the hard work is done (see Figure 1 in the paper). Following the definitions, $f =w_{q(x)}$ and $q(x)$ maps an instance to a leaf node. Putting this together gives:
$$
f = \begin{cases} 
      2 &amp; \mbox{age} &lt; 15 \mbox{ and } \mbox{male} \\
      0.1 &amp; \mbox{age} &lt; 15 \mbox{ and } \mbox{not male} \\
      -1 &amp; \mbox{age} \geq 15 
   \end{cases}
$$