Look for the comment in the article, after passing to a subsequence if necessary. The ultrafilter produces the necessary subsequence for any question that you ask, and will do so in such a way as to produce logically consistent answers for any combination of questions that you choose.

That is why the ultrafilter axiom is a weak version of choice. Take the set of possible yes/no questions that we can ask as predicates, such that each answer shows up infinitely often. The ultrafilter results in an arbitrary yet consistent set of choices of yes/no for each predicate.