The image below depicts a hypersphere inscribed in a hypercube. The number of dimensions increases from left to right (2,3,4 and 5 dim).
As you can see, the higher the dimension, the higher is the number of "hypercorners" that are left uncovered by a single hypersphere sitting at the center, of diameter no greater than the side. These increase as the powers of two: 4, 8, 16 and so on.
Hypersphere inscribed in hypercube
It is easy to prove, by using the Gamma function, that the relative volume of the sphere to that of the hypercube (aka "density") decreases as the number of dimensions rises.
In dimension 8, you would have 2^8=256 hypercorners around the 8-dimensional sphere.
One first trivial attempt to pack non-overlapping spheres in a fractal manner is by means of an Apollonian (Leibniz) gasket - thus observe the second picture attached. As the number of dimensions increases, so does the amount of spheres that need to be fitted in this manner, in an exponential manner.
Apollonian gasket (WP)
Of course, there are more efficient ways to pack spheres than this gasket. The simplest approximation is to think of unit hyperspheres: that is, of those having equal -unitary- volume. Or having equal surface area, in some other interpretations, as it is easy to translate from one to another (by using the pi formula).
Some more complex packing gaskets or schemes involve spheres of varying sizes, other than the classic Apollonian fractal, and their computation may involve optimization methods such as Linear Programming or meta/heuristics.
Viazovska's article does not deal with these, but only with unit balls, and benchmarks any packing method with the trivial classic lattice packing in 8 dimensions. Its conclusion is that there is none more efficient in density measures than the (E8) lattice.