This notation says $\mathcal{G}_n$ is a set whose elements are exactly the sets of the form $A\times \{H,T\}^\infty$ where $A$ is a subset of the set $\{H,T\}^n$. Let's unpack this a bit. An element of $\{H,T\}^n$ is just a sequence of $n$ letters, each of which is $H$ or $T$. So a subset $A$ is a set of such sequences.
The expression $A\times \{H,T\}^\infty$ literally denotes the set of ordered pairs $(x,y)$ where $x$ is an element of $A$ and $y$ is an element of $\{H,T\}^\infty$, i.e. an infinite sequence of $H$s and $T$s. However, in this context, this is not quite meant literally; instead, we are thinking of such an $(x,y)$ as representing a single infinite sequence of $H$s and $T$s obtained by concatenating $x$ and $y$ together. In this way, $A\times\{H,T\}^\infty$ is a subset of $\Omega=\{H,T\}^\infty$: specifically, it is the set of infinite sequences such that their first $n$ letters form a sequence which is an element of $A$.
For instance, suppose $n=2$ and $A=\{HH,TT\}$. Then $A\times\{H,T\}^\infty$ is the set of sequences that can be written as a concatenation of either $HH$ or $TT$ with some other infinite sequence. That is, it is the set of sequences that start with either $HH$ or $TT$.
Putting it all toghether, $\mathcal{G}_n$ is the set of all sets of the form $A\times\{H,T\}^\infty$, where $A$ is allowed to be any subset of $\{H,T\}^n$. Every element of $\mathcal{G}_n$ is a subset of $\Omega$, and there are $2^{2^n}$ elements of $\mathcal{G}_n$, one for each subset $A\subset \{H,T\}^n$. Another way to think of this is that $\mathcal{G}_n$ is the set of all subsets $S$ of $\Omega$ such that given an element $x$ of $\Omega$ (i.e., an infinite sequence), you only have to look at the first $n$ letters of $x$ to find out whether $x$ is in $S$. If $S$ has this property, then $S$ will be $A\times \{H,T\}^\infty$, where $A$ is the set of all sequences of $n$ letters such that if $x$ starts with such a sequence, $x$ is in $S$.
