Proving that L(G) is the language defined by the CFG G
Let us denote $L'$ the language defined by the CFG $S$.

To prove $L'$ = $L$ you must do two things:


prove $w' \in L' \Rrightarrow w'\in L(G)$, meaning $L' \subseteq L(G)$
prove $w \in L(G)\Rrightarrow w\in L'$, meaning $L(G) \subseteq L'$


Combine proofs 1 and 2, and you get that $L' = L(G)$.

The proofs themselves are not hard, since $L'$ has a very specific structure.