James L. A. Webb's Tropical Pioneers actually gets into this a fair bit for the West African sudanic grasslands. The short answer? Endemic disease that would wipe them out. It took quite some centuries to breed a horse resistant to parasites (eg., trypanosomiasis) and other organisms (later African Horse Sickness generally) that could see a natural increase. The exceptions fell outside of tropical areas (such as the horse-riding cultures of the far south, who acquired horses primarily from Europeans in the 17th and 18th centuries and became quite proficient, like the baSotho whom /u/huktonfonix points to).

That said, horses were actually present south of the Sahara in significant numbers because of the trans-Saharan trade. They formed the backbone of the feared Mali and then Songhay cavalry, as well as the cavalry of the Oyo Empire. Arabian horses were highly prized, but as I mentioned above, they had a very short lifespan south of the desert and so needed to be replaced regularly (thus the interest in developing a relatively resistant breed, eventually realized close to the Niger bend, albeit a much less impressive horse). In fact, the vectors for a variety of diseases are put off by high-contrast striped patterns, which potentially explains the existence of the zebra's coat. The zebra however isn't highly bred like a domestic horse is, for human ridership.

TL,DR: The disease ecology wasn't kind to equine physiology.

The peoples and empires of the Sahel zone (an area of dry grassland between the Sahara desert and the tropical rainforests of central Africa) did have horses. South of the Sahel zone though, the tse tse fly, among other tropical pests, kills horses readily which largely limited them to the Sahel zone until Europeans brought them by ship to the south were temperate climates kept pests to lower levels.

For more information on cavalry using empires of the Sahel, look up the Mali empire, Songhai Empire, and Kanem-Bornu for starters.

The Basotho do. Here's the breed that developed in Lesotho.