1. You'll need to play through World 1 normally
2. Once you get to the World 2, exit to the menu, go to Adventure Mode, and replay the entirety of World 1.
3. Once you do that, go back to Adventure Mode and you'll see that all levels up to the eighth in World 2 are unlocked.
4. You can repeat this process once you get into World 3, World 4 and so on.