Start with the plug end of the string of lights. Fold the plug cord in the fold of the cardboard and wrap the cord around the spool by grasping the ends and turning end-over-end (this prevents twisting the cords). Depending on the initial size of the cardboard and the length of the light strands, you should be able to get at least two strands on a spool. Plug the cord into its self to keep it from unraveling.
All done. Now box these guys up for next year.