Succinctly put, this says that the cartesian product of a set of nonempty sets is itself nonempty. Of course this sounds trivial.

The problem is when you're taking the product of a large number of sets, and you have no clear understanding of their structure. For instance, if you must choose an element of the cartesian product of an infinite number of copies of {0,1}, you can just choose (0,0,...,0,...), and you don't need to invoke the axiom of choice. But say you're trying to get a well ordering of the set of real numbers in the interval [0,1]. Initially, you have no problems: you could pick 0 as the first element, then 1 as the second (i.e. the first element of the rest), then maybe 0.5, followed by 0.3, 0.8, etc.

It's easy even to well-order some countable subset, placing them all at the beginning. But you then need to well-order a new subset, and you'll need to repeat this using some kind of transfinite induction (although you won't be calling it that, because you need the well ordering to do that). But we don't really know any ways of describing what happens at a non-countable "stage" of this process. The problem, in this case, is the repeated application of choice, not just any single application.

You could even make a case that the problem is with the Axiom of Infinity. Sure, it gives you the existence of an infinite set, but now you're looking at the structure of (all of) its subsets. And that's something we've no idea about!