In A Book of Set Theory by Charles C. Pinter, he states that:
"By an indexed family of classes, $\{A_i\}_{i\in I}$, we mean a graph $G$ whose domain is $I$; for each $i \in I$ we define by $A_i = \{x : (i,x) \in G\}$.
For example, consider $\{A_i\}_{i\in I}$ where $I = \{1,2\}$, $A_1 = \{a,b\}$, and $A_2 = \{c,d\}$. Then, formally, $\{A_i\}_{i\in I}$ is the graph
$G = \{(1,a),(1,b),(2,c),(2,d)\}$."
So, what I'm getting from this is that an indexed family of classes (or sets) is literally the graph from $I$ to $A_i$.
But when, you write it out as $\{A_i\}_{i\in I} = \{A_i : i \in I\}$ to me it seems like there's a list of classes $A_1,A_2,\ldots, A_n$, that are just simply elements of $\{A_i\}_{i\in I}$.
So in the example before, I would think $\{A_i\}_{i\in I} = \{A_1,A_2\}$ = $\{\{a,b\},\{c,d\}\}$, rather than it equaling G.
Best Answer
In an indexed family, we want to be able to recover the index. For example, given two indexed families $\{A_i\}_{i\in I}$ and $\{B_i\}_{i\in I}$ with the same index set, we may want to be able to speak of $\{A_i\cup B_i\}_{i\in I}$. If you write $\{\{a,b\},\{c,d\}\}$, the information which set is "first" is lost, i.e., $\{\{a,b\},\{c,d\}\}=\{\{c,d\},\{a,b\}\}$. Similar problems arise when $A_i=A_j$ may happen for some $i\ne j$.
For example, I almost get angry when people want to define linear independence of vectors for sets of vectors instead of indexed families of vectors. Only the family-based definition makes the statement "A square matrix is nonsingular if and only if the (set or family?) of its column vectors is linearly independent" true.
Apart from the above, if the $A_i$ are explicitly allowed to be proper classes, writing $A_i$ as element of something should make you frown, whereas the $G$ can at least make good sense as (then also proper) class.