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.
The below chart shows a canter/gallop animation speed comparison between a horse with 0 Strength and a horse with 100 Strength. Both have a Balanced (0) temperament:
Strength will reduce the stamina drained when galloping, but not when jumping. Below is a chart showing how long a horse can sustain the gallop at varying Strength levels:
Strength will increase the rate at which a horse regenerates their stamina when not performing an action like galloping or jumping. Below is a chart showing the impact Strength has on stamina regeneration at the long canter on a balanced (0) temperament (Keep in mind that a calm temperament, as well as travel speed, affect stamina regeneration! More info on this in the Temperaments and Basics section.):
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 below chart)
How likely your horse is to save your striding (see below chart) -- Thank you to Jessi for providing these figures!