The spheres are spikey, too, in the sense he means. Fix a small radius and consider balls of that radius centered on the surface of the sphere as you increase dimension. The volume of such neighborhoods that intersects the original sphere drops drastically. Just consider the cone that has its apex at your point on the sphere and that extends to the one lower dimensional sphere at the intersection of the sphere and your local ball. Even if your ball is so small that the cone has an apex angle of 89 degrees, that means that there is some fraction (less than one) of the cone base radius that you're losing. And that reduces your volume in each dimension. So there is exponential loss of cone volume as the number of dimensions increases.