Backyards can be purchased later in the game by tapping the green sign above your ranch. Each backyard starts with three horse slots. Backyard horse slots only cost 15 cash instead of the normal 20. To move things and horses into the backyard, select edit, then to the left highlight "send to backyard". Horses in your backyard cannot train or interact with other users.
Backyard 1 costs 20,000 gold.
Backyard 2 costs 500,000 gold.
Backyard 3 costs 5,000,000
Your backyard has a breeding stable. With this, you can select two horses in your backyard and breed them together. This costs the carrot value of the two horses combined, but no energy. While two horses are breeding in the backyard, they cannot be moved.
Horses in the backyard cannot breed with horses from other users ranches. A horse in the main yard that has bred with another users horse can be placed in the backyard while waiting on the foal.
The only way to get rare double-winged 72-hour ponies is by breeding in the backyard.