The most important property of generating functions is that their multiplication is translated to a convolution of the corresponding sequences:
If
$$f(x)=\sum_{n=0}^\infty a_nx^n$$ $$g(x)=\sum_{n=0}^\infty b_nx^n$$
Then $h(x)=f(x)g(x)$ can be written as:
$h(x)=\sum_{n=0}^\infty (\sum_{k=0}^n a_ib_{n-i})x^n$
The sum in the middle - $\sum_{k=0}^n a_ib_{n-i}$ is what I call "convolution". It comes from the rules of multiplying power series (very similar to multiplying polynomials).
Now, if $A=\sum_{n=0}^\infty a_nx^n$, then by deriving "term-term" we have:
$$A^\prime = \sum_{n=1}^\infty a_nnx^{n-1}$$
And so:
$$2xA^\prime = \sum_{n=1}^\infty 2na_nx^x$$
And so
$$2xA^\prime-A = \sum_{n=1}^\infty (2n-1)a_nx^x$$
And the result follows.
If you're new to generating function - congratulations! This is a fascinating subject which at first glance might seem intimidating, but is actually very elegant and beautiful. I suggest Wilf's "generatingfunctionology" or Stanley's "Enumerative Combinatorics", or Flajolet and Sedgewick's "Analytic Combinatorics" (both the letter books can be quite a challenge, but are worth it).