At first this is just one big finite
well, with expected spaced-out energies and corresponding wave
functions. Then seven walls are added, giving in essence eight
little finite wells, with progressively thicker walls between
them. The result: The previous levels divide into two "bands",
each of eight states.
What's the point? The eight little wells are like positive ions fixed in the regular lattice of a solid material, in the sense that each positve ion in a lattice attracts any wandering electron, meaning a lower potential energy when ion and electron are close, and the wandering electrons should thus be in a region of relatively high potential energy when halfway between the ions. So, the centers of the eight little finite wells are eight positive ions (low U) and there is a high potential energy (the high-U walls) between them. Okay, ions are not finite wells, but while changing the shape of the potential energies around each ion might be more realistic, the important qualitative result is the same: The allowed energies aren't spread out, but rather they group into bands, with as many allowed energy levels in each band as there are atoms in the lattice.
The animation suggests that the width of the bands depends
intimitely on the atomic separation. Good! For this is quite true!