To breed your own horses (mare and stallion) or breed with other player's stallions (stud) you must visit the Stud Farm.
Studding your horse to other players will earn you income from the stud fee. The amount of the stud fee will vary depending on the rating of the horse.
Studding with your own horses incurs a vet fee. Similarly to the stud fee, the vet fee will also vary depending on the rating of the horse but will be less than a stud fee for using another player's similar rating horse.
Breeding outside of your stable does have advantages. It will most likely diversify the offspring and potentially change the inheritable traits. If you breed only within your stable, you may find only minor changes in the offspring you produce, and breeding the same horses twice will produce similar results with only some variations.
Horses will pass on various traits to their offspring including, track likes and dislikes, fatigue recovery, maximum/minimum strength and endurance, training stats and of course, their rating. Each breeding will produce different results, even if they are with the same mare and stallion. Breeding also may increase the offspring's rating (if you are lucky!).
The stud fee is paid at the time of breeding (i.e. when you select the horse from the stud farm).
There are two reasons why the breeding can appear to go 'backwards' in strength or endurance.