To unlock a new world, you will first need to complete all the levels in the current world you are in. Each land will have 10 or 20 levels associated with it. The final required level you must defeat will have a locked gate after it. Once you beat the level, the gate will unlock and the next world will appear. To view a list of all of the worlds you have unlocked and fast travel to them, tap the trophy icon on the bottom right, and then tap the "Map" button.