Class World will open up automatically as you progress through the story. If you don't know what it is it's still too early yet. Basically Class World powers up your characters permanently. You can take a character in there to improve various attributes or to learn moves from other characters. For story characters this is the only way to learn more advanced weapon techniques and spells. You'll need to have a fist specialist learn Big Bang first. Then you can use Class World to transfer it to Mao or Laharl.