It's sort of like a neopets thing. you grab the eggs from the site. They need different people to vie them for them to mature and they need a certain amount of click throughs. Adult dragons need views once every eight weeks, so i occationally cycle them through my sig to prevent accident.
The most interesting things I have so far are a rock dragon, magi, and some mysterious purple egg. I like adopting them rather than just going for the parentless cave ones, it seems cooler to me and I can look at other people's dragons in the process.
The click think is why I individually click all your eggs once every day if they are on the board.
If your egg has soft shell or your hatchling is sick, it generally means your egg/hatchling is getting too much exposure. Go to your scroll. Pick the "actions" option next to the sick egg and choose "hide." Leave the egg/hatchling hidden overnight, then unhide to see if they are fine again.
Oh, yea - I heard (but idk if it's true) that if the overall views ratio to the unique views is more than 10:1 the egg/hatchling will die. Maybe if you get close to that the dragon gets sick or the shell gets soft.