The story is a retelling of a problem in set theory. In set theory, we have sets, collections of objects. Some sets contain infinitely many elements, as the set of all even numbers. A set can contain sets. It also contains the empty set, by the way.
All sets contain the empty set.
We can define the set of all sets. Some sets contain themselves. The set of all red marbles does not contain itself, because it contains no sets at all, only marbles. The set of all sets obviously contains itself. Well, let's construct a very interesting set, the set of all sets which do not contain themselves .