Following Probability and measure Theory by Ash (2000).
let $\Omega$ be a set, let $C$ be a class of subsets of $\Omega$ and $A \subset \Omega$, we denote by $C \cap A$ the class $\{ B \cap A : B \in C \} $. And the minimal sigma field over $C$ is denoted by $\sigma(C) = F$.
Ash wants to show that $B \cap A \in \sigma_A (C \cap A)$.
Let $L$ consist of those sets $B \in F$ s.t. $B \cap A \in \sigma_A (C \cap A)$.
Now Ash states that $C \subset L$. Why is this true?
Edit with original text:
You've omitted a lot of the text in you initial question -- perhaps you overlooked something there? And you also seem to be unfamiliar with the script font: $\mathscr S$ is a script S, not an L. (L would be $\mathscr L$).
Anyway, to the question: Why is $\mathscr C\subseteq \mathscr S$? Let's consider some $B\in\mathscr C$ and show that it is also in $\mathscr S$. For this we have to show that (1) $B\in \mathscr F$ and (2) $B\cap A\in\sigma_A(\mathscr C\cap A)$.
(1) $B\in \mathscr C$, and $\mathscr F=\sigma_\Omega(\mathscr C)$ by definition contains $\mathscr C$ as a subset. So $B\in\mathscr F$, as desired.
(2) Since $B\in \mathscr C$ we have $B\cap A\in\mathscr C\cap A$. By definition of $\sigma$, $\mathscr C\cap A$ is a subset of $\sigma_A(\mathscr C\cap A)$, so $B\cap A\in\sigma_A(\mathscr C\cap A)$, as desired.