The meaning of Generating functions

The meaning of Generating functions

I had a look to this video on the field of series and sequences which I
know not much about!
This guy looked for the generating function of $a_n = 2a_{n-1} + 4a_{n-2}$
with $a_0=1$ and $a_1=3$. The generating function is
$$A(x)=\frac{1+x}{1-2x-4x^2}$$ (solution showed at 04:33 in the video)
I've been trying to understand what is the real meaning of this generating
function he found with wikipedia but I got lost into thousands of pages
and definitions.
Can we find with this formula the 18th number of the series? What does the
x represents? How does this generating function "represents" the series?
Hope my question is not too broad and that you will easily find where is
my gap!