What does $F:2^\nu \rightarrow \mathbb{R}$ mean?

This is indeed a function that maps subsets of $\nu$ to real numbers.

So, why is this notation used?

In set theory, if we have two sets $A$ and $B$, the notation $A^B$ (or ${}^BA$ by some authors) commonly denotes the set of all functions from $B$ to $A$, that is each function $f:B\to A$ is a member of $A^B$, and nothing else is.

Another common thing in set theory, is that the natural number $n$ is a shorthand for the set $\{0,1,\dots,n-1\}$. In particular, $2$ denotes the set $\{0,1\}$.

So what is $2^\nu$?

Well, it's the set of functions $f:\nu\to\{0,1\}$. Each such function $f$ uniquely describes a subset $X\subseteq\nu$ in the following way: if $a$ is an element of $\nu$, then we say that $a\in X$ if and only if $f(a)=1$.

Conversely, each subset $X\subseteq\nu$ uniquely describes a function $f:\nu \to \{0,1\}$, where we let $f$ send an element $a\in \nu$ to $1$ if and only if $a\in X$, and send it to $0$ otherwise.

Such function $f$ is called the characteristic function of the subset $X$. You could consider $f$ and $X$ represent the same concept in two different ways.

Thus, in conclusion, $F:2^\nu\to \Bbb R$ is a map of the set of characteristic functions (hence essentially of subsets of $\nu$) to real numbers.