The notation $[n]$ is an $n$-element set of integers. Usually, it means $[n] = \{1,2,\ldots,n\}$, but sometimes it goes from 0 to $n-1$ (this will basically never matter, and just assume it’s the first one).
In general, $2^{S}$ could mean either the powerset of $S$ (i.e., all the subsets of $S$), or could mean the set of functions mapping from $S$ to the set $\{0,1\}$. People use these interchangeably via characteristic functions as in your quote.
People also call $2^{[n]}$ the $n$-cube or the Boolean cube. It’s just the collection of subsets of $[n]$.
You’ll know what folks mean by the surrounding notation (if ever unsure).
For instance, $A \in 2^{[n]}$ would suggest that $A$ is a set (since it’s a capital letter near the beginning of the alphabet), so in this context, $2^{[n]}$ seems to be the set of subsets of $[n]$.
On the other hand, if someone wrote $\forall f \in 2^{[n]}$, I would say this must be referring to the collection of functions. But this second notation is sort of odd.
Let me spice it up once more!
Say $\Omega = \{0,1\}^n$ (now there is no ambiguity since the exponent is a number. This must be the collection of length $n$ strings consisting of $0$’s and $1$’s) Then $\Omega$ is also called the Boolean cube and there’s a natural correspondence between these vectors, subsets of $[n]$ and functions from $[n]$ to $\{0,1\}$. The corresponding vector is called the “characteristic vector” of the corresponding set.
Then we can say things like $f : \Omega \to \{0,1\}$, which is saying $f$ is a function that takes 01-strings of length $n$ and outputs a value of 0 or 1. (I.e., $f$ is a “boolean function”).
Good luck with it all. You’ll get the hang of it soon enough, I’m sure!
Added:
By the way, looking at your quote, I’m guessing boolean functions were defined as mapping from strings (aka vectors) to $\{0,1\}$, i.e., maps from $\Omega$ (as above) to $\{0,1\}$. And the author is using $2^{[n]}$ to mean the collection of subsets of $[n]$. So the author’s point is that instead of viewing the inputs as strings, you could view them as sets, and in that way you could say boolean functions take in a subsets of $[n]$, and they spit out a number.
Example:
Say $n=3$, and $f(x_1 x_2 x_3) = x_3$ [this is called a dictator function, since the output is determined by only one bit of the input string (they like social choice metaphors)].
Then a string like $011$ corresponds to the set $\{2,3\}$ (where the $011$ can be interpreted as describing a set one element at a time as “it’s missing 1, it contains 2, it contains 3”).
Then viewed a function on sets the same map $f$ would be $f(S) = \begin{cases} 1, \qquad \text{if $3 \in S$}\\ 0, \qquad \text{otherwise}\end{cases}$.
