MagiStream

I've seen and clicked on people who have more than 4 eggs how do they manage that?

You can have a maximum of five non-Donation eggs, and two additional eggs if I remember correctly, if they're Donation creatures.

Also, if people have unopened gifts from a gifting event left over and open them they'll end up with a lot of eggs.
Also there's a couple quests that reward you with eggs that'll give them to you regardless on if you have space or not (the easy beginning quest that gives 4 wasps comes to mind, even if you're egg locked it'll drop 4 eggs on you and it totally within the rules)