I also found it very weird, but here's my intuition.
There are 2^k > 512 spheres stuck to eachother across k-d (pretend k=9). The line from the center to the point where the inner sphere touches one of outer spheres has to shortcut through all k dimensions to get from the center to the sphere.
This distance has been massively inflated due to the number of dimensions. But the distance to the edge of the box hasn't been inflated - it's just constant, so the inner sphere breaks out.
There are 2^k > 512 spheres stuck to eachother across k-d (pretend k=9). The line from the center to the point where the inner sphere touches one of outer spheres has to shortcut through all k dimensions to get from the center to the sphere.
This distance has been massively inflated due to the number of dimensions. But the distance to the edge of the box hasn't been inflated - it's just constant, so the inner sphere breaks out.