All explanations I found on the axiom of choice seem to say more or less the following: Although the axiom of choice is "intuitively obvious," it is necessary because making infinitely many choices is not "built into" the first order logic.
I'm not so familiar with the first order logic, but which part of the following causes problem with the first order logic?
Let $X_1,X_2,X_3,\ldots$ be a sequence of nonempty sets.
For each $X_i$, there exists an element $x_i\in X_i$.
Let $f$ be a function on the set $\{1,2,3,\cdots\}$ defined by $f(i)=x_i$.
All statements look "first order" to me.
