To breed horses, you need to have a male and a female horse on your farm. You'll also need a breeding station, which can be purchased from the shop at level 6. To breed your horses, simply click on your breeding station, select the button with the house, and then tap on one of the free breeding slots with the bottle. Drag your male and female horses into the hearts and click on the checkmark below the horses on the left to pay the breeding fee.
You'll see a selection of three possible offspring. You can click on the checkmark to pay Rubies to receive the one of your choice, or you can click the button with the dice at the bottom to receive a random foal (free). Your foal will initially be kept at the breeding station, but soon it will be old enough to move into a real stable.