The hoof has to fulfill two important functions. It needs to carry the weight of the horse, and it needs to balance the horse during locomotion, when at some points in time, the horses' weight is carried by a single hoof.
The illustrations below are referenced from this very informative website.
The static load is distributes equally on both front legs:
dynamic load however can be concentrated on either one of the two front legs:
In order to balance the weight of the front half of the horse on a single hoof, the horse loads the lateral side 10-20% more:
I have found, that the horse performs this "balancing act" by moving the leg slightly towards the mid-line.
It is interesting to note that the same conclusion has been drawn by the author of this website:
The article is in German, but the first picture shows this specific way of advancing the legs towards the midline.
Taken together these observations suggest (as has been done here) that the lateral side of the horse's hoof is wider and flatter because it is the "balancing side" of the hoof, and the medial side is primarily involved for the static load bearing of the leg.
It may also be worth noting that as the weight of the horse increases, so is the need to balance the horse's weight during locomotion, when temporarily only one leg carries the weight. The heavier the horse, the more balancing on the lateral hoof side is required.