"A standard saddle designed for comfort during long journeys. Its durable material holds strong in all sorts of weather, which has made it a favorite among travelers."
Breath of the Wild
The Traveler's Saddle can be obtained by scanning a Link (Rider) amiibo, through the use of the amiibo Rune. Scanning the amiibo will drop various items from the sky, along with a treasure chest, which has a chance of containing the Traveler's Saddle, but can also contain different items. The amiibo can only be scanned once per day.
Just like all horse gear items in the game, this gear part does not provide protection for the horse, and it serves mainly cosmetic purposes.