The following proof is taken from Introduction to Topological Manifolds by John Lee. There is another question on this proof but it does not address my query.
I would like to ask about the third sentence. Since $e\subseteq Y$ is a cell, by definition of the subcomplex, $Y$ contains the closure $\overline{e}$. The 'C' in CW implies that the closure of $e$ intersects finitely many cells (since it is contained in the union of finitely many cells). But I'm unsure as to why this implies that the cells of $X$ that $\overline{e}$ intersects has to be cells of $Y$ too?
This has not too much to do with topological issues; it is more just a set theoretical consequence of a few facts about $Y$ that follow from the definition of subcomplex.
By definition of subcomplex, $Y$ is a union of cells of $X$. And since the cells of $X$ are pairwise disjoint, for each cell $e'$ of $X$ it follows that either $e' \cap Y = \emptyset$ or $e' \subset Y$.
So for each cell $e'$ of $X$, if there exists $x \in e'$ such that $x \in Y$ then it follows that $e' \subset Y$.
For each cell $e$ of $X$ such that $e \subset Y$ and for each $x \in \overline{e}$, as you say it follows from the definition of a subcomplex that $x \in Y$. Letting $e'$ be the unique cell of $X$ such that $x \in e'$, it then follows from the argument above that $e' \subset Y$.