Eggs drop from fights and from the breeding pens. They wait in the soul bag until you hatch them into creatures.
MENU > SOUL BAG holds your creatures and your eggs (1). HATCH ALL (2) shows how many eggs are waiting.
Click an egg and confirm (1): the creature inside is revealed and joins your soul bag.
From floor 8 HATCH ALL (1) opens every egg in one go, egg after egg on the stage; before that, each egg is hatched by hand.
Screenshots are taken from the live game by tools/tutorial_shots.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: eggs, system-breeding.