Full Idea
For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice.
Gist of Idea
We can only define functions if Choice tells us which items are involved
Source
Herbert B. Enderton (Elements of Set Theory [1977], 3:48)
Book Reference
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.48