Animal breeding will be unlocked when you reach level 14.

After a certain number of feeds, Adult animals become Mature animals, and a house bubble will appear above them. Tap on it to make them go into the Breeding Station.

Mature animals of the same breed in the Breeding Station will produce Baby animals and unlock new breeds.

The building will have a heart bubble above it once it has a pair of Mature animals that are ready to breed.

The Breeding Station has 3 tabs:

Breeding - Shows a list of your available pairs of Mature animals that are ready to breed.

You can breed your Mature animals here by tapping the “Breed” button.
It also shows the Tier of the breed that you will get from breeding your Mature animals. You can get items faster from higher Tier animals because their feeding count is higher than the previous Tier!
The Animals that have a house bubble are Mature animals that are still in their respective animal homes. You can call them into the Breeding Station to let them breed.

Babies - You can view the details of the Baby animals that you have in this tab. Using Energy, you can speed up their growth into Adult animals.

Mature - Shows you the Mature animals that you have in your Breeding Station. You can also breed them directly by selecting a pair and then tapping the “Breed” button.

Once you have successfully bred your pair of Mature animals, a Stork will deliver a Baby animal in a basket.

Tap the Star bubble to collect your XP and wait for the Baby animal to grow.

You can speed up the Baby's growth by recruiting a Farmhand and using Energy.

When the baby animal is fully grown, a green check mark will appear over its head. Tapping on it will turn the Baby into an Adult animal. After that, its Mature parents will become Elder animals and will automatically be sold for Elder Animal Points. To learn more, please visit: What are Elder Animal Points?

Note: You will receive XP as a reward for unlocking new breeds.