Badges are a way to display your specific accomplishments or milestones achieved within Hatchlings. You can earn a badge for your Hatchlings level, an impressive Challenge Streak, by participating in in-game events, or by finding select eggs of extreme difficulty (the Kingly series is a good example of this).
How are they earned?
The badges are awarded to you within the game. Hatchlings will notify you when you've won a badge, and it will be displayed on your Basket Profile under "Badges Achieved."
What does one do with them?
Badges are passive displays of your accomplishments in Hatchlings. Be proud!
Why can't I earn this badge anymore?
Not all badges can be earned at all times. A good example of this is the Make-A-Wish Foundation Badge. Some badges can only be earned during a temporary promotion or charitable-donation drive (such as the Make-A-Wish Badge). The badges themselves never expire (they'll always be displayed on your Basket Profile), but sometimes they can only be achieved during a specific time-frame.