Axiom of Choice : Is it Axiom? or Trivial Fact? This question might be rude or inappropriate, but I am very beginner of studying mathematics.

Today I had learned the concept of Axiom of Choice, which states that $
\forall$ nonempty indexed set $S_i$, there $\exists$ indexed family of element $x_i$ where $\forall x_i \in Si$. 

To me, at first it reached out to me as a just mere fact, but can't understand why this axiom must have been accepted as an axiom. 

It's little confusing to me.

To wrap up the notion of this OP, it could be summarized as below:


Why does the concept of axiom of choice must be accepted as axiom?
What makes a statement to be valued to be designated as an axiom?
In "naive" set theory, sets are collections of things. In "axiomatic" set theory, sets are whatever satisfies the axioms -- abstract mathematical objects. Just as chess pieces need not be pieces of wood that are moved on a board sitting on a table, but may be bits of information in a computer, so also sets need not be sets of the kind you were told about in seventh grade.

Similarly in geometry, points and lines need not be points and line in physical space but can be whatever satisfies the axioms. And one finds that certain axioms are satisfied by things that don't satisfy the parallel postulate. Likewise the axioms of set theory other than the axiom of choice can be satisfied by things that don't satisfy the axiom of choice.

Is there are "set of all left socks" when infinitely many pairs of socks exist and in each pair there is no objective way to single out which one is to be the "left" sock? Some people object on philosophical grounds to the idea that there can be a set containing one element of each pair when there is no rule to say in each case which one belongs to that set.
In "naive" set theory, sets are collections of things. In "axiomatic" set theory, sets are whatever satisfies the axioms -- abstract mathematical objects. Just as chess pieces need not be pieces of wood that are moved on a board sitting on a table, but may be bits of information in a computer, so also sets need not be sets of the kind you were told about in seventh grade.

Similarly in geometry, points and lines need not be points and line in physical space but can be whatever satisfies the axioms. And one finds that certain axioms are satisfied by things that don't satisfy the parallel postulate. Likewise the axioms of set theory other than the axiom of choice can be satisfied by things that don't satisfy the axiom of choice.

Is there are "set of all left socks" when infinitely many pairs of socks exist and in each pair there is no objective way to single out which one is to be the "left" sock? Some people object on philosophical grounds to the idea that there can be a set containing one element of each pair when there is no rule to say in each case which one belongs to that set.