There's a (sort of) intuitive way of understanding why things like these are possible without even seeing it visually.
How can you make two solid 2D squares occupy the space? They can't travel through each other, so...? Easy. Lift one of the squares into the 3rd Dimension, move it to the same X,Y coords, then drop it on the other (think of putting stacking 2 pieces of paper).
Similarly, you can have two 3D objects occupy the same space by "lifting" one into the 4th dimension, moving it to the same X,Y,Z coords as the other, and then "dropping it". You can "cheat" by thinking of the 4th dimension as time - one object travels through time, places itself at the same position of where the other object is, then returns. But as I said, that's "cheating".
I do similar mental exercise with higher dimensions, but for me with easier situation - escaping from certain area without crossing boundaries.
1 dimensional - an endless line with 2 spots marking boundaries to some subpart X. If you're inside, you need to enter 2nd dimension to escape that subpart X without crossing boundaries (and ie come back to original dimension).
2 dimensional - circle, you're inside, you need to enter 3rd dimension to escape without touching the circle.
3 dimensional - sphere, entering 4th dimension to escape without touching the sphere.
I can't go with my simpleton mind much further but it should scale indefinitely :)
How can you make two solid 2D squares occupy the space? They can't travel through each other, so...? Easy. Lift one of the squares into the 3rd Dimension, move it to the same X,Y coords, then drop it on the other (think of putting stacking 2 pieces of paper).
Similarly, you can have two 3D objects occupy the same space by "lifting" one into the 4th dimension, moving it to the same X,Y,Z coords as the other, and then "dropping it". You can "cheat" by thinking of the 4th dimension as time - one object travels through time, places itself at the same position of where the other object is, then returns. But as I said, that's "cheating".