I am sure it is to do with the military, with cavalry officers carrying swords strapped to their left leg, so lifting the leg with the sword would be difficult so you lift the right leg and swing it over the horses back.
I agree it could be a military thing.
But it is not a good idea as the horse only gets used to being mounted from the left side. The horse might be a bit wary if you potter around it's right side, be it mounting or whatever.
I am not sure about this question and possibly all the other answers could be right. But think about the people who ride bikes, as a right handed person I find it very difficult to mount my bike from the Right Hand side, so I wonder whether this also has something to do with balance when you are mounting a horse. Perhaps one or two Left handed people will tell us which side they would mount from on either Horse or Bike. Cheers. ALLIGATOR