An explicative definition of what is meant by $\{A_i\}_{i\in I}$? What does $\{A_i\}_{i\in I}$ mean exactly?
I know it's an index, but what exactly is that?

It denotes a collection $\mathscr A$ of objects, and it means there is surjection $f: I\rightarrow \mathscr A$. One often has that $f$ is injective, so that objects $f(i) = A_i\in \mathscr A$ and $f(j)= A_j\in\mathscr A$  corresponding to distinct indices $i\in I$ and $j\in I$ are themselves distinct, but this need not be the case.
It simply means that there are sufficiently many members of $I$ to enumerate the members of $\mathscr A$.
