Existence in variants of the Axiom of Choice
Think about it this way: you're trying to write down instructions for a robot to pick an element from each set. The instructions have to be completely unambiguous.
If you just say, "OK, each set $A_i$ has an element $a_i$; so let $f(i)$ be that element," this clearly doesn't unambiguously tell the robot how to pick some element from each $A_i$. Suppose $A_i=\{x, y\}$ - which one is $a_i$?
On the other hand, suppose you had a well-ordering $W$ of the union of the $A_i$s lying around. You could tell the robot:
On input $i$,
go over to $W$,
and look for the $W$-least element of $A_i$.
Pick that one.
This is unambiguous! It's dependent on $W$, of course - different $W$s will give different procedures - but as long as you have at least one $W$, you can write this algorithm. To drive this home: if $A_i=\{x, y\}$, then I don't know which of $x$ or $y$ is $a_i$, but I will know as soon as I look at $W$ and see whether $x<_Wy$ or $y<_Wx$. Think of a well-ordering as a kind of disambiguator.
The main issue here is defining concretely, from the existing universe, a choice function (or whatever object).
I have two white socks, plain, new, entirely indistinguishable. I hold one sock in my left hand, and the other in my right. Now I turn around for a few seconds and turn back to you. Did I switch the socks?
You can't tell. Your power of observation made it possible to observe there are two socks, but not to discern them in any meaningful way. So you don't know if I switched them.
So now, if I have infinitely many such pairs, you cannot tell me, in finite time, how to choose one from each pair. Because you have to go to each pair separately, and pick a sock (and mark it so I won't fool you later).
Similarly here. The existence of a well ordering allows us to uniformly label all the socks, all the elements. But when we only know that the sets we want to choose from are nonempty, that is not enough to describe---in the mathematical universe, with out "current" tools---what would be the choice function.
A good visual way to 'see' the problem is this:
For the first method, imagine each well-order of $\cup_{i\in I}A_i$ is a marble, all of which are put into a hat which is the hat of well-orders of that set. We then invoke the well-ordering principle to show that there is at least one marble in that hat, and then reach into that hat and take out a marble (any marble, doesn't matter which one). From here, we use that ordering to pick all our $a_i$'s.
For the second method, instead, for each $i\in I$, represent the $a_i$'s in $A_i$ as marble's in the hat labelled $A_i$ Now we are needing to take a marble from each of the $|I|$ hats we have to construct our choice function, and this is where the problem is: now we have to make many choices, in fact, if we don't have a well-order for $I$, then there is no way to come up with a procedure that we can guarantee will remove a marble from every hat in $I$ (said marbles will be the $f(i)$'s that define our choice function).
Do you see the problem now?