Proof for multiplying generating functions

Casebash is correct that this is a definition and not a theorem. But the motivation from 3.48 (Defintion of product of series) of little Rudin may convince you that this is a good definition:
$\sum_{n=0}^{\inf} a_n z^n \cdot \sum_{n=0}^{\inf} b_n z^n = (a_0+a_1z+a_2z^2+ \cdots)(b_0+b_1z+b_2z^2+ \cdots)$
$=a_0b_0+(a_0b_1 + a_1b_0)z + (a_0b_2+a_1b_1+a_2b_0)z^2 + \cdots$
$=c_0+c_1z+c_2z^2+ \cdots $
where $c_n=\sum_{k=0}^n a_k b_{n-k}$

It is actually the other way round. A generating function is generally defined to have an addition operation where the components are added and a multiplication operation like that you mentioned. Once we have made these definitions, we observe that polynomials obey the same laws and so that it is convenient to represent generating functions as infinite polynomials rather than just an infinite tuple.