In many cases you are "translating" the higher-dimensional geometry into something that is not geometric or which is much lower dimensional. You don't generally visualize 24 dimensions. You can get a decent intuition for 4 with practice but at some point this breaks down.

For example, the 24-dimensional packing corresponds to the Leech lattice which itself corresponds to the Golay code:

https://en.wikipedia.org/wiki/Leech_lattice

https://en.wikipedia.org/wiki/Binary_Golay_code