Horsemen have traditionally worked from the left side of their horses—standing at the horse’s left side to put on a halter, bridle, or saddle, mounting the horse on the left side, etc. The left side is called the near side; the right side is called the off side.
This custom had its roots in practicality. Most people are right-handed, so it was easier or more natural to hold onto a horse’s halter or lead rope with the right hand. This puts the horse handler on the horse’s left side.