To start off, you need to find a Chao Box in one of the levels. Break open the box and you will find a Chao Key. If you keep this key, then when you finish the level, you will be warped to Chao World!
While you look around the Chao Garden, you should see two Eggs. These are the Chao Eggs! Now, how to hatch them...
Hatching The Egg
Before you can do anything in the Chao world, you have to hatch your Chao egg. Without doing this you won't get anywhere! There are three ways of hatching the eggs, each way having different effects.
Shaking the egg is the most common way of hatching an egg. It usually produces a nice, mild Chao. Pick up the egg by holding the 'B' button, while still holding it, move the analouge pad to shake it. After about 5-10 seconds of shaking the egg, place it back down. The egg will hatch giving you a Chao!
You can also throw the Chao egg against a wall to hatch it. To do this, pick up the egg by pressing the 'B' button, then while moving, press 'B' again. This will not make your Chao very happy, but it produces a more active, lively Chao.
Your Chao egg will hatch naturally after a while. Simply leave your Chao egg alone and it will hatch by itself. Hatching this way will produce a normal Chao