It's easy to construct, just build a point, extrude the point out to make a line, extrude the line out to make a square, extrude the square out to make a cube, then extrude the cube out to make a hypercube. It looks like two cubes with the corners connected by lines through a 4th dimension that we can't construct in 3D.
One way to visualize it is to use time, so if you translate a cube from point A to point B, the "extrusion" would be along time and you can kind of visualize that extra "dimension". But what would scaling, rotation, etc along that invisible axis look like? To fully grok it, we'd need to be able to sculpt the hypercube's shadow in 3D as easily as drawing a cube on a piece of paper.
Asking in the hopes of building a mental bridge to 4D and then possibly 5D in order to generalize to higher dimensions.