Although we can't know for sure, we have a good answer for this one.
We know that the modern horse, Equus caballus was descended from a wild horse, Equus ferus (now extinct) that existed all across Asia and Europe.
We know that most of the male lines of the modern horse came from somewhere in central Asia - with a wide variety of female lines that suggested people were augmenting their herds by catching wild mares.
Archaeological evidence also points this way. For example, in Botai, in northern Kazakhstan, we have evidence of established horse domesticating dating back to at least 3100 B.C. The evidence? Skeletal changes caused by being ridden, marks on the teeth that clearly came from bits, and traces of horse milk on pottery.
Horses were, thus, first domesticated by the people of the central Asian steppes, who used them (and still use them) for transportation, meat, and even dairy. (I'll address why we don't often use horses for dairy in another post).
For worldbuilding purposes, you could easily pin the domestication of horses down to any steppe or plains-related human group. Horse riding was probably invented as a way to go faster than your own legs could carry you in open areas.
This Icelandic mare represents a primitive and isolated breed and is probably very close to the first European domestic horses in size and build.