Why are CFLs not closed under intersection?
Let us assume $2$ CFLs $L_1$ and $L_2$ and their corresponding grammars be $S_1$ and $S_2$ respectively.
It is very straightforward to see that the union of the two, represented by the new grammar as
$$S \to S_1 \mid S_2$$
is also a CFG, as the rule of being context-free is still not violated. Context-free grammar

But to prove that they are not closed under intersection, I'll provide an example.
Let $L_1$ and $L_2$ respectively be:
$$L_1 = \{ a^nb^nc^m \mid n, m \ge 0 \}$$
$$L_2 = \{ a^mb^nc^n \mid n, m \ge 0 \}.$$
It is not hard to see that $L_1 \cap L_2$ is
$$L = \{ a^nb^nc^n \mid n, m \ge 0 \},$$
which is not CFL (No PDA cannot accept that language).
Hope this explains.