The simplest expression of this property is this: "There are six objects constructed in a canonical way on a set of 6 points. 6 is the only number for which this is possible." This is pretty vague so I'll quote the theorem:
Six is the only natural number n for which there is a construction of n isomorphic objects an an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the points of A.
Pretty amazing result! Except, embarassingly, I'm a bit confused about this because I'm not sure what "naturally" means here. Come to think of it, I'm not sure what "isomorphic" means here either. I don't remember ever doing a course on "objects" and their isomorphisms. The authors make the meaning of "natural" clearer over the page in a category-theoretical definition:
the category whose objects are the n-element sets, and whose morphisms are the bijections between them has a non-trivial functor to itself if and only if n=6
Which is all very well except I don't know what "non-trivial" means here. Or at least I think I know what it means and I can construct counterexamples. My brain must be working sub-par today. Ho hum.
Anyway, there is one version of this property whose statement is absolutely unambiguous to me: the permutation group Sn has an outer automorphism only for n=6. In fact S6 acts on the set of 6 objects in two inequivalent ways. Interestingly a bunch of 'exceptional' constructions follow from this: the Hoffman-Singleton graph, the projective plane of order 4, the Steiner system S(5,6,12).
This reminds me a little some other outer automorphism that also leads to an exceptional object: the "triality" outer automorphisms of Spin(8) which lead to a well known exceptional object - the octonions.
Hmmm...I just discovered that John Baez has written his own musings on the number six.