Before the Great Calamity, it was apparently a riding course located near Hateno Village in between Fir's River, Camphor Pond, and Lake Jarrah for horseback riding and horseback archery. Presumably it was used by Hyrulean Soldiers and Knights of Hyrule stationed at nearby Fort Hateno for horseback combat training before the fall of Hyrule Castle.
One hundred years after the Great Calamity, the ruins of the riding course became a camp for Bokoblins and Moblins. However despite this the Sign on the fork in the road leading to both the course and Hateno Village has not been changed as it still reads "Equestrian Riding Course". Though the area serves little purpose other than an enemy camp, the course is still littered with archery targets and they can still be used by Link to practice horseback archery, provided he eliminates all enemies in the area. Additionally, Link can obtain a Serpentine Spear, Shield of the Mind's Eye, Emblazoned Shield, and Wooden Shield that spawn inside the camp where the Bokoblins and Moblins may grab and wield them if they are alerted to Link's presence. Additionally a Korok can be found hiding here and there is a Raft on the shore of Lake Jarrah.
There is also an abandoned log cabin that uses the same model as the Woodcutter's House though unlike the Old Man's cabin on the Great Plateau its bed is broken and unusable, though it can still be used as a shelter where Link can create a campfire when it is raining or take shelter during a thunderstorm. The cabin also lacks a Torch, though like the Woodcutter's House some items can be found inside the cabin.
Despite the courses name appearing on the sign, its name does not appear on the Sheikah Slate map. Interestingly, it is implied by Jini at the Mounted Archery Camp that Horseback Archery was apparently invented here as he mentions it being invented out east somewhere and is the only place in the east fits the description of a Horseback Archery range. However it should be noted that Horseback Archery has existed since Ocarina of Time where it was practiced by the Gerudo and Hero of Time. Additionally, both Hero of Twilight and Zelda practice Horseback Archery in Twilight Princess as do Bulblins. However it is possible that knowledge of the art was lost then rediscovered at the Equestrian Riding Course and Jini mistakenly assumes it was invented there as both the events of Ocarina of Time and Twilight Princess have faded into legend by the time of Breath of the Wild.