If $S$ is a non-empty set, then we can take $r$ such that $r \in S$.
Suppose that for any $a \in A$, $S_a$ is a non-empty set.
Then, we can take $r(a)$ such that $r(a) \in S_a$ for any $a \in A$.
Why do we need the axiom of choice in 2.?
What is the difference between 1. and 2.?
Do I need to study basics of axiomatic set theory to appreciate the axiom of choice?
I used to have the same question as the OP. I've seen quite some different explanations, including the famous "socks" example but I am not quite convinced by any of them.
My own interpretation, in the more intuitive, informal language, is that any formal, rigorous mathematical reasoning needs to be done in finite steps; the limitation of such reasoning prevents us from going from 1. to 2. without introducing another axiom, e.g., the Axiom of Choice.
In fact, for the same reason, we can construct every finite ordinal from the empty set and the Axiom of Pair, but not $\omega$, which is infinite: each finite ordinal can be constructed in finite steps, while $\omega$ cannot without an extra axiom, e.g., the Axiom of Infinity for the existence of $\omega$:
