How does ammonium chloride increase the solubility of magnesium hydroxide in water?
There is a strong reason. But it is the chemical reaction, not just a better dissolution.

$\ce{Mg(OH)2}$ is a base with the limited solubility, defined by $K_\mathrm{sp}=[\ce{Mg^2+}][\ce{OH-}]^2$

$\ce{NH4+}$ ion, created by $\ce{NH4Cl}$ dissolution, acts as a  weak acid:

$$\ce{NH4+ + H2O &lt;&lt;=&gt; NH3 + H3O+}$$

with $\mathrm{p}K_\mathrm{a}=9.25$

$\ce{OH-}$ ions formed by dissolution of $\ce{Mg(OH)2}$ are eliminated by recombination

$$\ce{OH- + H3O+ &lt;=&gt;&gt; 2 H2O}$$ 

what supports the dissolution by keeping the product of ion concentrations below the $K_\mathrm{sp}$.

Effectively, there is ongoing equilibrium:

$$\ce{Mg(OH)2 v + 2 NH4+ &lt;=&gt; Mg^2+ + 2 NH3 + 2 H2O}$$