From Mac Lane's Category Theory:
Let $C$ be a category and let $X$ be a discrete set and let $A = \text{C}^X$ be the functor category.
Then $A$ has as its objects the $X$-indexed families $a = \{a_x : x \in X\}$ of objects of $C$.
I'm having a difficult time seeing how the objects of $A$ are $x$-indexed families of $C$.
If a functor $(F : X \rightarrow \text{C})$ is an object of $A$, then it's defined entirely on where it sends each $x \in X$. So I can see how an object of $A$ should be a collection of objects of $C$ that are defined by the particular functor $F$, i.e., an object of $A$ should be $a_F = \{c \in C : F(x) = c\ \text{for some $x \in X$}\}$.
But I'm not seeing how the objects are $x$ indexed families.
Can someone explain what I'm missing in the analogy here?
You are confusing the functor $F$ and its image in $C$.
Just like a function $f:X\to Y$ between sets is not the set $\{y\in Y\,|\, f(x)=y \text{ for some } x\in X\}$, a functor $F:X\to C$ is not $\{c\in C\,|\, F(x)=c \text{ for some } x\in X\}$.
Rather, $F:X\to C$ is the association of some $c\in C$ for each $x\in X$, so we may write $c_x\in C$ for this particular element, and this gives a family $\{c_x\in C\,|\, x\in X\}$. The fact that $X$ is discrete as a category implies that the functor is determined by just this family (there is no need to worry about the action on morphisms).