Strength has two distinct roles:
Directly increases horse animation speed over the flat, which results in faster ground coverage. Note that because Strength works by increasing animation speed, the horse will cycle through their canter/gallop animation more quickly and will have a shorter jump window. For more information on the stride and jump window, please see The Stride page.
Increases stamina regeneration at rest (not jumping or galloping) and reduces stamina drained at the gallop.
Strength will reduce the stamina drained when galloping, but not when jumping.
Strength will increase the rate at which a horse regenerates their stamina when not performing an action like galloping or jumping. Keep in mind that a Calm temperament also affects stamina regeneration (more on this on the Temperaments page)!:
Agility simply improves a horse's turning radius. There just isn't much else to say here.
The Jumping skill affects three things:
The maximum height your horse can jump:
Low: 0+
Medium: 30+
Tall: 70+
How much stamina is consumed when jumping (see Figure 4.)
How likely your horse is to save your striding (see Figure 5.) -- Thank you to Jessi for providing these values!