In fact you are given two languages
- $L_1$ defined as a set of strings of balanced parentheses.
- $L_2$ defined as a set of strings with equal number of ('s and )'s and every prefix of w contains at least as many ('s as )'
You have to prove that these two languages/sets are equal. One way to prove it is to demonstrate that the grammar you indicated in your post does generate both languages.
Another way is to prove $L_1 \subset L_2$ and $L_2 \subset L_1$.