The $>>0$ just means for all sufficiently large natural numbers. Now how to swallow this confusing definition. Intuitively a polynomial in $R[x]$ is just $a_0 + a_1x + ... + a_n^n$ for some $a_0,a_1,...,a_n \in R$. Of course, this is not a very rigorous definition. So let's analyze this definition. First of all, $f(n)$ is supposed to represent the coefficient on $x^n$ (or maybe $x^{n-1}$ depending on what the convention for $\mathbb{N}$ is in Denmark). So our polynomial is just $P(x) = \sum_{n \ge 0} f(n)x^n$.
The $f(n) = 0, n >> 0$ just means for all sufficiently large $n$ we have $f(n) = 0$. This is to guarantee the polynomial has a bounded degree. So this is basically how this definition works. Its essentially defining the polynomial as a sequence of its coefficients, where for all sufficiently large indices we have the element equals $0$.