The breeding chance determines if the breeding will be successful or not. Therefore, you want your breeding chance to be fairly high so that there will be offspring.
If you do get lucky with your breeding, you can get anywhere from one to six baby animals. However, your breeding chance has no influence on this.
You can increase your breeding chance in a number of ways:
Placing Pools and Hideouts in the Pair's Pen - A Pool and a Hideout will increase a pair's breeding chance by 5% and 10% respectively.
Unlocking Gold Breeding Slots - Every time you unlock a golden breeding slot, your breeding chance will be increased by 5% for all breedings in your store.
There are a total of 3 golden breeding slots, so you can increase your chances by up to 15% by unlocking these.
Buying Fertility Food - Fertility Food guarantees the success of a breeding (that is, makes it 100%) and can be purchased for Diamonds.