Why did the kings carry out …

It was used by ancient Indian kings to prove their imperial sovereignty: a horse accompanied by the king's warriors would be released to wander for a period of one year. In the territory traversed by the horse, any rival could dispute the king's authority by challenging the warriors accompanying it. After one year, if no enemy had managed to kill or capture the horse, the animal would be guided back to the king's capital. It would be then sacrificed, and the king would be declared as an undisputed sovereign.

The ritual is recorded as being held by many ancient rulers, but apparently only by two in the last thousand years. The most recent ritual was in 1741, the second one held by Maharajah Jai Singh II of Jaipur. The original Vedic religion had evidently included many animal sacrifices, as had the various folk religions of India. Brahminical Hinduism had evolved opposing animal sacrifices, which have not been the norm in most forms of Hinduism for many centuries. The great prestige and political role of the Ashvamedha perhaps kept it alive for longer.