The Sleeping Forest is an area located just behind Bone Village on the northern continent. The only way to reach this area is by going through Bone Village.
The story leads the team on their way through the Sleeping Forest when they are tracking down Aeris. This is after they have visited the Temple of the Ancients and just before they reach the Forgotten Capital. Check out the Bone Village walkthrough page for more details.
The only way to get through the Sleeping Forest is by obtaining the Lunar Harp (check the walkthrough page listed above for details). Otherwise the screens will just repeat as you work your way north and you can never actually get through.
You can obtain the Kjata Summon Materia in this area. It can be found floating around in the trees - you just have to predict where it is going to appear and grab it.
The map below shows the location of the Sleeping Forest (which can be found be entering Bone Village) on the World Map: