As you progress trough the Hogwarts Legacy story, you’ll get a chance to breed beast. You’ll be able to feature by competing Deek's side quests. During his mission "Foal of the Dead" new pens will be unlocked in the Room of Requirement and you can summon a breeding pen.
Breeding a beast is as simple as releasing a male and a female of a species into the same pen where there is a breeding pen. Then use the sign next to the enclosure and confirm that you want to breed offspring. Note that you can only breed offspring of the same species. Crosses between different species are not possible.
You then have to wait 30 minutes in real time for the offspring to appear in the enclosure. You can then care for and raise them like all other animals.
How to sell beasts?
You can also earn money with beasts. You can sell beasts at the Brood and Peck shop in Hogsmeade. For each type, regardless of the rarity, you get 120 gold pieces here.