Consider a vector of stochastic variables $Z = (Z_1,...,Z_J)$. We assume each $Z_j$ is Frechet distributed
$$Z_j \sim F(z_j) = \exp(-z_j^{-\theta}),$$
and that they are mutually independent such that $F_Z(z) = \prod_{j=1}^J \exp(- z_j^{-\theta}) = \exp(-\sum_j z_j^{-\theta})$. Furthermore, it is known that
$$ \mathbb E[Z_j] = k(\theta) = \Gamma\left( 1-\frac{1}{\theta} \right).$$
Given this set up we have the following properties (left unproven here):
When you scale $Z_j$ with a constant $A_j$ the distribution becomes $A_jZ_j \sim F(z) = \exp(-A_j^{\theta}Z_j^{-\theta}).$
The probability that $i = \arg \max_j \{A_jZ_j\}$ is given as
$$\pi_i = \frac{A_i^\theta}{\sum_j A_j^\theta }.$$
We will use these properties for deriving equation (10) in the text you are reading. First we define the utility function og agent $h$ for alternative $j$ as
$$U_{jh}=Z_{jh}\frac{W_{j}Q_{j}}{P_{j}^\beta},$$
and we notice that the scaling constant $A_j = {W_{j}Q_{j}/P_{j}^\beta}$. This then implies that the probability $\pi_i$ of an agent choosing city $i$ is given as
$$\pi_i = \frac{\left( {W_{i}Q_{i}/P_{i}^\beta}\right)^\theta}{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta },$$
using that the labour force size is exogenously given the size of the labour force in city $i$ must be
$$ L_i = \pi_i L = \frac{\left( {W_{i}Q_{i}/P_{i}^\beta}\right)^\theta}{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta } L \Leftrightarrow \\[8pt]
(\star) \ \ \frac{P_i^\beta L_i^{1/\theta}}{Q_i} \underbrace{\left( \frac{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta }{L} \right)^{1/\theta} }_{\text{utility per worker}} = W_i,$$
where the utility per worker is simply a scaling constant independent of $i$ the index of the city under consideration.
How can we see it is utility per worker? The agent choose the alternative that provides max utility. Define the max utility
$$ \hat U_i = \max_j \{U_{j}\},$$
then the $Pr(\hat U_i \leq r) = Pr(\text{all} \ U_j \leq r) = \prod_j Pr(U_j \leq r)$ using independence. However
$$\prod_j Pr(U_j \leq r) = \prod_j \exp(-A_j^\theta r^{-\theta}) = \exp\left(-r^{-\theta} \cdot \sum_j A_j^\theta\right),$$
which is seen to be a Frechet distribution and define $s:=\left(\sum_j A_j^\theta\right)^{1/\theta}$ we can rewrite it to get
$$\prod_j Pr(U_j \leq r) = \exp\left(-r^{-\theta} \cdot s^\theta\right) = \exp\left(-(r/s)^{-\theta}\right),$$
for which the expectation according to wiki is
$$\mathbb E[\hat U] = sk(\theta) = \left(\sum_j A_j^\theta\right)^{1/\theta}k(\theta) = \left(\sum_j \left( W_{j}Q_{j}/P_{j}^\beta\right)^\theta\right)^{1/\theta}k(\theta),$$
if you insert this in equation $(\star)$ you have utility per worker except that workers are $L^{1/\theta}$ - since $L$ is exogenous variable this is without consequence.