To start a burrow, you must dig. Peck will show you how to. Press the icon with the shovel and pickaxe. Two options will show up. The options are Dig and Fill. If you press Dig, you can plan your burrow as underground is sorted into squares. Press the squares and make a room. When all the squares you have chosen are green, you can press the green Dig button. Your bunnies that are unoccupied will start to dig. Digging takes up a lot of energy, so you will have to feed your bunnies when they run out of energy. The Fill button can be used to undo the squares you dig. It does not take any bunnies, coins, or stars. Just press the previously dug squares to cover the dirt.
There are sections of dirt. Every five levels, you will unlock a new section of the underground to dig. You can tell which section you are seeing by the color. At the bottom is a lava lake.