The Legend of Zelda: Ocarina of Time introduced Epona the horse to the series, and they were created for a specific purpose. Epona is a recurring Zelda character that appears in multiple games, serving as Link's steed, yet she was an optional character in her debut appearance. Ocarina of Time represented a huge jump forward in technology from the previous entries in the series and Epona was created to fulfill a specific role in the game.
Ocarina of Time introduced big open environments that took a long time to cross. The most notorious of these was Hyrule Field, which could be a pain to sprint across on foot, especially at nighttime, when Link could be attacked by Stalchildren that crawl their way out of the ground. If the player makes their way to the Lon Lon Ranch and completes the Epona quest (which involves activities in both the past and present), then they can use Epona's Song to summon their horse and ride her across Hyrule. Epona was a returning Zelda: Ocarina of Time character in Majora's Mask, and they have made sporadic appearances since.
The 8-bit and 16-bit entries in The Legend of Zelda franchise didn't require a mount, as the hardware of the time could only handle small-scale areas. Ocarina of Time had a much grander scope than A Link to the Past and Link's Awakening, which is why Link needed a new mode of transportation. Ocarina of Time producer Shigeru Miyamoto discussed the reasoning behind Epona's inclusion in an interview printed in The Legend of Zelda: Ocarina of Time Encyclopedia. The interview was translated by Shmuplations and the full piece can be read on their website, along with other rare interviews given by Japanese game developers.
Miyamoto - The scenes with Epona were the result of long discussions with Yoshiaki Koizumi (who was also responsible for Link's design this time). The reason we chose a horse is because we wanted the vehicle, or mode of transportation, to be something that brought you into contact with others. That choice is also a reflection of the staff's understanding of what "Zelda-ness" means, to be sure.
Zelda: Ocarina Of Time - Epona Brought Link Into Contact With Others
According to Miyamoto, a horse was chosen to create a mode of transportation that would bring Link into contact with others. This could refer to Zelda: Ocarina of Time's extensive quest surrounding Epona, its connections to the characters in the setting, or the fact that it makes it easy to engage with NPCs while mounted. It makes sense that characters in the setting and the player would grow more connected to an animal than an inanimate object, like if the Master Cycle from Breath of the Wild had been used instead of a horse.
There were plenty of times when The Legend of Zelda series couldn't include Epona in its story, such as The Wind Waker flooding Hyrule and requiring boats for travel, or the technology advancing in Spirit Tracks to the point where steam trains became available. Fans had to wait over a decade to saddle up Epona, as she appeared in Twilight Princess in 2006 and would make her next playable appearance in 2017. Despite these gaps in time between appearances, Epona is a popular Legend of Zelda supporting character, and many fans are eagerly awaiting her return in the sequel to The Legend of Zelda: Breath of the Wild.
Source: The Legend of Zelda: Ocarina of Time Encyclopedia, Shmuplations