The Cartesian product of non-empty sets is itself non-empty.
The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple with one element drawn from each of the sets being, um, Cartesianly multiplied.
Thus, the Cartesian product of the three sets {1, 4}, {a, b}, and {@, 2} is the set {(1,a,@), (1,a,2), (1,b,@), (1,b,2), (4,a,@), (4,a,2), (4,b,@), (4,b,2)}.
The Cartesian product of the three sets {1, 4}, {}, and {@, 2} is {}, the empty set, because no tuples exist such that the second element of the tuple belongs to the set {} (the second Cartesian factor).
So all the Axiom of Choice asserts is that, if all of the Cartesian factors are nonempty, then a tuple exists with one element drawn from each of the Cartesian factors. The only way for it to be impossible for such a tuple to exist is if one of the factors itself has no elements.
It's a theorem for finite Cartesian products, so all the dispute is over infinite products.
It's probably also worth mentioning that the C in ZFC stands for the Axiom of Choice, which is an indicator that people have explored not using it. ZFC without the Axiom of Choice is ZF.
