As it was my first tutorial I explained some things in complicated way :)
I said to parent each empty to each edge- I meant locate the empty to each edge,
which will be used to rotate around. Empties are parented to each other and only
the last 4th empty IS A PARENT of the cube( cube is a child ).
Sorry for the mess.
I also used a lot of shortcuts without mentioning about them.
Anyway I hope I made it clear enough to understand the principle.

If you would like to make the cube rotate more times than just 4 you have to use
the same principle with just more empties. For example if you want the cube to
rotate 8 times create 8 empties, locate them on the edges( each edge two empties
next to each other and call them in the way spiral goes- from 1 to 8 )
then parent cube to the last 8th empty, the 8th empty to the 7th etc.

Here's link to the tutorial which inspired me to make my version
of this project:
http://www.creativebloq.com/roll-object-its-edge-blender-1114198