When your egg is ready you will see a glowing circle form around it. Tap this circle to reveal your new caterpillar!
From here, you can tap on the caterpillar to move them to a cosy nook of your forest where you can drag and feed them leaves from above. They will require multiple feedings that take varying amounts of time, but once they are fully fed they will turn into a chrysalis/pupa.
If you want there is the option to watch videos to hurry your caterpillar! This is accessed by tapping the caterpillar and then tapping the play button. Each video can take 30 mins off of a caterpillar or chrysalis of your choosing. Remember that video availability can vary and there may not always be an available video!