In some restaurants, they are forced to. They would probably prefer to dine casually unshod, but rules are rules.
They don’t need to, but they heard horseshoes are lucky.
They are to protect the horse’s hooves from excessive wear or even damage. A lame horse is a dead horse.
It sets them apart from the uncouth, unshod, wild horses.
Detailed here: https://horseyhooves.com/why-do-horses-need-shoes/#:~:text=1%20Regular%20Shoe.%20The%20regular%20shoe%20is%20made,eight%20weeks%20by%20an%20experienced%20and%20knowledgeable%20farrier.
The domesticated, human owned horses do.