Once you have bought a second stable on your ranch, you can transfer horses from your main stable to your second stable. First, go to your stable by clicking on the Stable icon on the left. Then, click on the "Your stable" tab at the top of the page (next to the star icon), and click on "Transfer horses". Select which horses you want to move.
Note: The horses in your second stable count towards your total horse number, but they will not show up when you try to view your horses from your homepage. To see your horses in your second stable, you must go to your second stable either by clicking on the icon on your ranch, or by using the same tab as "Your stable"; just click "Second stable" instead of transfer horses.