So first of all- this is a rumor. There is no reason other than personal preference/ surroundings/ etc. to decide which side to mount on. So Let's look at where the idea came from that mounting a horse "should" be done on the Left side.
Now think about getting on. If you have a sword hanging on the left side of your body, would it not be easier to mount from the left, as to not have to get the sword to the opposite side of the horse? It is logical to assume, that Left-handed knights would choose to mount from the other side, as the sword would be resting on the Right side of their belt.
Luckily, horses have grown drastically in size since medieval times, but it was still a task to be had. In the Greek general Xenophon's book "The Art of Horsemanship", he outlines instruction on how to mount one's horse. Assuming the rider carried his sword on his left hip, he explains that the rider is to grasp the mane up by the ears of the horse with his left hand, and down by the withers with his right... And swing up. You can probably imagine how difficult this would be if you were to have to somehow kick the sword end over the horse's back in the process.
So, as with many things in life, the idea that mounting "should" be done from the left is simply a matter of tradition. In reality, it is safe and practical to have a both the horse and rider comfortable with mounting and dismounting from both sides, so if a situation should arise where only one side of the horse is practical (often found on trail rides), your horse won't be learning a new trick!