I am currently reading about the independence of the continuum hypothesis from ZFC following the topos theoretic proof given in Chapter VI.3 of MacLane Moerdijk's Sheaves in Geometry and Logic.
The goal is to construct a new model of set theory $\mathscr{S}'$, in which CH fails. For this we need a set $B'$ right between the set of natural numbers $\mathbb N'$ and $\mathcal{P}'(\mathbb N')$. We assume that our given set theory $\mathscr{S}$ does not admit such a set (otherwise there is nothing to prove). To construct $\mathscr{S}'$ we realize the required injections by means of a presheaf topos $\mathsf{Psh}(\mathbf{P})$ on the Cohen-poset $\mathbf{P}$ of finite approximations. We then make this presheaf topos into a boolean topos in the universal way by passing to the Cohen-topos $\mathscr{S}'=\mathsf{Sh}(\mathbf{P},\neg\neg)$. So far so good.
It is the final step I fail to properly motivate. To make the argument work we need the canonical functor $\mathscr{S} \rightarrow \mathsf{Sh}(\mathbf{P},\neg\neg)$ sending a set $X$ to the constant sheaf $\hat{X}$ to preserve strict cardinal inequalities. This can be shown by means of the following theorem:
Let $\mathscr{E}$ be a Grothendieck topos generated by Souslin-objects (objects $X$ with the property that any family of pairwise disjoint subobjects of $X$ is at most countable). Then the canonical functor $\widehat{(-)}$ preserves strict cardinal inequalities.
While I can follow the proof given by MacLane Moerdijk, I don't understand the intuition behind this theorem. I tried to read about the Souslin-property, but I could not find much more on it than the fact that it originated in Souslin asking for an alternative characterization of the real numbers. I don't understand why it reappears in this particular form in the proof of the independence of CH. Phrased differently:
Why does the Souslin property appear in the theorem above? Is it a necessary consequence of $\widehat{(-)}$ preserving strict cardinal inequalities (hence being a characterization)? Or is it maybe obvious from the definition of the Cohen-poset that it having the Souslin property will have some strong consequences for sheaves on it?
I hope this question is specific enough to be on Math.SE, since I feel like it might have an obvious answer for someone with more experience in set/topos theory than myself. Anyway I thank you for your time and answers.