Functions don't have components, vectors (or points) have components.
The function $f$ maps elements of $U$ to points in $\mathbb R^n$. "The action of $f$ on input $x$" just means what $f$ does to $x$, i.e. the output $f(x)$, which is a point in $\mathbb R^n$. Maybe it'll be clearer if we give this point a name, $y=f(x)$. Then $y$ can be "written out in component form" as $(y_1,y_2,\ldots,y_n)$. We can go further and treat $y_1, y_2, \ldots, y_n$ themselves as $n$ different functions of $x$, namely $f_1,f_2,\ldots,f_n$ which each map an element of $U$ to a number in $\mathbb R$.
For example, suppose $f:\mathbb R\to\mathbb R^2$ is the function that maps $x$ to the point $(\cos x,\sin x)$. The two components of this point are $\cos x$ and $\sin x$. We can define two functions $f_1(x)=\cos x$ and $f_2(x)=\sin x$, and then say that $f(x)=(f_1(x),f_2(x))$.