Every linear mapping satisfies $f(0) = 0$ In my textbook is written that every linear mapping $ f: V \to V'$ satisfies $f(0)=0$. But what about the mappings between the polynomials like $p(x) = ax^2 + bx + c$, where $c$ is nonzero? Elementary probably, but fundamental.

You are getting confused between $f$ and $p$.
$f$ is a function that applies to the coefficients of the polynomial and returns new coefficients. 
For clarity, we can write
$$f(a,b,c)=(a',b',c')$$
and a necessary condition for linearity is
$$f(0,0,0)=(0,0,0).$$
For instance,
$$f(a,b,c)=(2a,b+c,c-a),$$ also understood as
$$f(ax^2+bx+c)=2ax^2+(b+c)x+(c-a)$$
may be linear because
$$f(0x^2+0x+0)=0x^2+0x+0.$$
This has nothing to do with the exaluation of the polynomial at $x=0$, and mandatorily involves a polynomial such that $c=0$.
