Eggs can be hatched into Bees and Queens by using the Hatching Machine in the game or by using the Hatching Machine website. 

Egg shards (which are combined into Eggs) can be obtained through various methods like Hunting for shards, Lucky Wheel, completing Quests, playing games in the Game Mall, owning Land, or purchasing them from the Item Marketplace via the shop.

You can also buy Eggs from the in-game Asset Marketplace via the shop.

NOTE: Learn more about Hatching Gen Z Bees here