Because if you need to "store" the information, as the author stated, you would need some kind of mechanism for doing so. It would be impossible to have such a mechanism that would be less than an atom in size, otherwise representing the set would be a superset of itself (ie: you can't represent more atoms than the universe contains if the cost of representing one atom is more than one atom).

In reality, what the author meant was "contains," which means you have a bunch of different bits with multiple simultaneous values (I believe this is distinct values in many different dimensions, but please correct me if I'm wrong), and the number of possible combinations of those values (since the combinations all exist "at once") can be more than the number of atoms in the universe. Hope that makes sense.