In medieval times, swords were the weapon of choice. Most people are right handed, and left-handedness was extremely uncommon back then. If you are right-handed, you pull the sword from the scabbard most easily if it's on the left side of your body. It's hard to swing your leg and a sword over a horse, so it became common to mount from the left side, because then you stick your right leg (the one without the sword next to it) over the horse to mount. The practice never really fell out of style, although many horses now are trained to mount from both sides, the left simply remains the more popular side.