The Wikipedia link cleared up some stuff, as did the dogfeathers animation. However I still find it odd that we say this thing "exists" physically yet it can't truly be seen. When I see a 3D object on my screen, I'm seeing 3D projected to 2D. Seeing 4D projected to 2D just doesn't do it justice. How can I see a 3D projection of a 4D object? In other words, a physical, tangible model of a hypercube. Is this possible?
I'm still not seeing how it's real. You build this hypercube thing with cubes the same way you build a cube with squares, starting with the classic "t" shape. But when you're done, it's still an object that can be measured in x, y, and z planes only. At least, that's how the 2D projection makes it seem.