“The egg will hatch naturally after a while. You can also help the egg to hatch. Depending on how you hatch the egg, the characteristics of the Chao will be affected.”
There are three ways of hatching Chao eggs, with each way having different effects on your Chao.
Your Chao egg will hatch naturally after a while, even with no interaction from you. Simply leave the egg alone and it will hatch by itself after a few minutes.
Your Chao’s appearance (including facial expression) and personality is decided when the egg is first formed. How you hatch the egg has no effect on on either of these things.
Shaking or rocking the egg is the most common way of hatching an egg. Pick up the egg using the action button, and while still holding the button down, move the analogue stick or directional keys to shake it. After about 5-10 seconds of shaking the egg, place it back down on the ground. The egg will start to bobble, and then your Chao will hatch from it.
You can also throw the egg to break it open. To do this, pick up the egg by pressing the action button, and then throw it by pressing the action button again while moving. If you throw the egg into open space it will not break open, you will need to throw it against a wall.
Your Chao will not look very pleased at first if you use this method, but there do not seem to be any negative effects from doing this.* All that will happen is that the Chao will start with an angry/upset expression above its head; however, this expression will only last a few moments, and the Chao seems perfectly happy afterwards.
Many people choose to hatch their eggs in this way because it is far quicker than the other two methods. Hatching the egg by throwing it will affect it’s kindness negatively, but will make it more energetic.
*In the Dreamcast verison of Sonic Adventure, hatching your egg by throwing it will cause your Chao to be very unhappy. It will cry and will even hiss at you if you come near, but it will not lose any HP. Cheer it up by feeding it!
The help orb in the garden will tell you that the way you hatch the egg affects its personality. A lot of sources say that this method produces a lively Chao, but this is because of the lowest mood status shown in Chao Adventure.
After the Chao has hatched from the egg, the top half of the eggshell will remain behind as an interactive item. You can pick this item up and and give it to your Chao, or sell it for rings at the Black Market. If you give it to your Chao it may wear it as a hat, or throw it away uninterested. For more details about Chao hats please see this page on the Chao Wiki. Sonic Adventure 1 on the Dreamcast does not have Chao hats, and therefore will not leave an eggshell behind upon hatching.