How exactly do I catch wild horses?

[circusseal]

I've been searching and searching, and no one seems to explain it. At what point do wild horses begin to spawn? I haven't seen any at all.

[SinisterScarecrow]

You need to do a certain number of quests before you are able to buy different Wild Horse Pens from an NPC. After you purchased one, or multiple, each day you play afterwards has a CHANCE to spawn wild horses, but it is not a given.
You will get a notification if the NPC sighted Wilds somewhere, it also mentions on which map.

Make sure you have free space in your stable, because once you step onto the map you CANNOT LEAVE unless you want the horses to despawn/run away (so you'll loose that herd and need to wait for another).

If there's anything unclear still, I'm happy to help whereever I can. :)