The movie shows the stereographic projection of a rotating four dimensional cube onto a three dimensional hyperplane. Imagine the cube as a wire frame in three dimensions. The picture you see on the screen is an additional projection onto a two-dimensional plane.
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. [Source: Wikipedia]