Another second for Flatland.

If your world was confined to a line where there was only forward and back, could you conceive of an up, down, left, or right? (Don't cheat and use your 3-D mindset, really imagine you are in a 1-D world.) So if you can't think of 2-D/3-D in a 1-D world, does that mean 2-D and 3-D universes are impossible? Obviously not. Same thing for beyond 3-D spatially. There is no way you could "feel" it or have a mental grasp of it.

You could imagine that color represents the 4-D spatial dimension. Something can be specified spatially in 3 dimensions, and its value in the 4th dimension is represented by assigning a color. This would allow you to "see" a 4-D surface as a topography in 3-D space. Just like a 2-D plot with color scale (or contour lines) can help you "see" a 3-D image.