The idea behind the axiom of choice is to tell you how to choose when you can't necessarily distinguish between the items.
When you take out a pair of socks from the closet (or drawer, etc) you can't tell which one you had on your left foot and which on your right, or which one is this and which one is that. If you can, well... you need to remember to let someone else make sure you have matching socks :-)
The idea, formally, is that if you take a product of infinitely many non-empty sets then it is not empty, namely there is a function in the product which returns an element in each coordinate.
Formally, given a set $I$ such that $\forall i\in I$ we have $A_i\not=\emptyset$ then there is $F\colon I\to\bigcup_{i\in I} A_i$ for which $F(i)\in A_i$.
If you think about it, this is not a "strange" requirement when you are discussing mathematics, and it might come naturally in many places.
For example, for any two sets $A,B$ if there is $f\colon A\to B$ which is surjective then there is an injective $g\colon B\to A$. What are we doing? In a sense we choose one representative in each equivalence class of $a_1\sim a_2\iff f(a_1) = f(a_2)$. But if we have infinitely many equivalence classes - then we (usually) need to invoke some choice axiom (perhaps the axiom of choice, or countable choice, or dependent choice, etc...)
However, many times the use is not of the axiom of choice, but rather an equivalent principle called "Zorn's Lemma", it states that if you have a partial order in which every chain is bounded from above - then there is some maximal element (that is no one is strictly above it in the order). Again not something that is unreasonable if you want infinitary processes to "act" similar to finitary ones.
Addendum:
After I gave it some extra thought, I came up with something that might clear up the things. There is a concept which is "a definable element", that is that you can write some formula $\psi(x)$ such that $a$ is the only element (suppose in $A$) which satisfies the formula. If you want to choose from infinitely many sets then you need to be able and tell which element you have chosen.
If you have at least one definable element in each set, suppose by some uniform formula $\varphi(x)$, then you can clearly choose the one element defined by $\varphi$. However if there are many definable elements in each set, or infinitely many formulae are needed - then you cannot express it simply, and then you must assume that you can do it.
And as I remarked above, we simply want infinitary processes to behave well, like finitary ones.