The short answer is that we don't really know; they are related to some of the most mysterious aspects of representation theory and number theory.

What we know is that these dimensions support lattice structures with exceptional symmetries. The intricate geometry of these symmetries interacts with number theory to provide spectral data (automorphic forms) that have very rare properties; this is the starting point for Maryna's investigations.

Okounkov's popular scientific exposition "The magic of 8 and 24", linked in the article, goes into considerable detail on the constructions.

They contain extremely many symmetries. The Leech Lattice is certainly a uniquely symmetrical object.