How to prove prove $L(G) = \{~w\in\{a,b\}^*~|~\#_aw= \#_bw\}$ for my CFG $G$?
There are two implications involved in the correctness. Every string in $L$ can be derived by the CFG (completeness) and every string derived by the grammar belongs to $L$ (consistency). I think you want an argument to formalize the first implication by induction.

Assuming the string $x\in L$ starts with an $a$, there must be a position with letter $b$ such that the string is of the form $aubv$ such that the $\#_a(u)=\#_b(u)$. The argument I know is by counting. Read the letters in $x$ one by one, and add $1$ for each $a$, subtract $1$ for each $b$. As the number of $a$'s equals the number of $b$'s, the count starts and stops with $0$. The counts becomes $1$ after the first letter, and must drop back to $0$ at some position with a $b$. In between we have seen an equal number of $a$'s and $b$'s. 

Now that $x=aubv$ with $x\in L$ and $u\in L$ it is easy to argue that also $v\in L$.