This is how I learnt it:
You can construct a 1-dimensional object (a line) by taking two points and joining them:
<pre>A • • B => A •--------• B</pre>
You can construct a 2-dimensional square by taking two parallel lines and joining them:
<pre>A •----• B => A •----• B
| |
C •----• D C •----• D</pre>
You can construct a 3-dimensional cube by taking two parallel squares and joining them in the same way. I'm not drawing that. :p
Finally, you can construct a 4-dimensional tesseract (or hypercube) by taking two parallel cubes and connecting each correlating point.
Since we only have three <i>visually observable</i> dimensions, most of the time the 4th dimension of 4-d hypercube is represented visually as permutations of the cube structure over time, which is why that picture you supplied is rotating like that.
I stole this image from Wikipedia and highlighted each cube so that you can see how each of the two cubes connect together.