For the sphere inside cube you can draw these out and get some intuitions about how the ratio of empty space changes.
One thing you can think out is how if you pick two random vectors in a high dimension they are almost certainly orthogonal. In 2D, pick a random vector. There will only be 2 vectors orthogonal to it, right? Now do the same in 3D. There's a whole plane orthogonal to your vector! That's a hell of a lot more than 2! Move up into 4D and you have a 3D-hyperplane that's orthogonal.
The spikes on the hypercube might have some visual intuition. In 2D you have 4 corners and you can imagine the smooth rounding off into a circle. 3D we now have 8 corners and the sphere again looks rounded off but you can see here how the hypercube gets spikey but whatever you think the hypersphere looks like you're likely wrong. Even the cube's intuition still fails you with this thinking so you need to be careful
