Speak to Zyle on the path through Gerudo Canyon to being the quest.
This isn't very difficult; Zyle wants to buy a horse for 300 rupees. If you ride a horse to this guy and give it to him, that will complete the quest. If you don't want to ride a horse all the way there, you can register one and then ride it from the Wasteland Stables. If you're coming from that direction, note you may have to move boulders with stasis to get your horse to him.
Make sure you don't sell him one you want to keep for yourself. You won't get it back.
You ran into a traveler named Zyle on a road in the Gerudo region—he's in trouble after losing his horse on his journey.
He says he'll give you 300 rupees if you bring him a horse...but it has to be appropriately sized for Zyle's body frame.
Zyle lost his horse during his travels and he seems to have approved of the one that you gave him to replace it.
You gave him the horse for 300 rupees.
After completing the quest, Zyle can be found again at Outskirt Stable. He has sold the horse you gave him.