You need 2 animals of the same breed living in the same Animal Home to produce baby animals and unlock new breeds.

When an animal is ready to breed, a Heart bubble will appear on top of the animal.

Tap the Heart bubble and a Stork will deliver a baby animal in a basket.

Tap the Envelope bubble to claim the baby animal.

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 to help you out. Tap on the baby animal and assign a Farmhand that will feed the baby animal to grow.

When the baby animal is fully grown, a green check mark will appear over its head.