There seems to be no logical explanation for this. If moss works remotely the same on Planetos as it would on Earth, it should grow on the north side.
Moss on which side?
On Earth, moss tends to grow on the north side of trees, because this is in the shadow. This prevents water from evaporating, generating a nice moist place for the moss to grow. The reason moss needs this is that it doesn't have roots like most other plants, so it can't suck up water from the ground.
Moss on south side implies the sun arches through the north, instead of through the south
In the Northern Hemisphere here on Earth, the sun passes from East to West, through the South. This causes the north side of trees to be in the shadow more than the southern side.
If moss grew the opposite direction, logically the sun should reflect that difference. So does it?
It does not
There are a multitude of quotes that the sun does indeed rise in the East and set in the West, but there is no direct information on whether it stands in the south at midday. However, we can judge the direction by the shadows cast by the sun:
The Night's Watch permitted the forest to come no closer than half a mile of the north face of the Wall. The thickets of ironwood and sentinel and oak that had once grown there had been harvested centuries ago, to create a broad swath of open ground through which no enemy could hope to pass unseen. Tyrion had heard that elsewhere along the Wall, between the three fortresses, the wildwood had come creeping back over the decades, that there were places where grey-green sentinels and pale white weirwoods had taken root in the shadow of the Wall itself, but Castle Black had a prodigious appetite for firewood, and here the forest was still kept at bay by the axes of the black brothers.
A Game of Thrones - Tyrion III
We have the Wall, which runs in a straight line from West to East (or East to West, depending how you look at things). If the sun stands in the south, the shadow is north of the wall, if the sun stands in the north, the shadow is south of the wall.
Given that there are trees growing in the shadow of the wall on the northern side, this shows that the sun stands in the south, just like ours.
Additionally, the opposite would imply that Castle Black would be almost permamently in the shadow of the wall. That sounds cold.
The sun on Planetos (is it really on Planetos?) follows a similar path as ours, so logically the moss should grow on the same side as on Earth.
It could be that somewhere in the middle of Westeros is the equator, meaning that south of it the moss would be the other way around. The quote from TheLethalCarrot would imply that for that to hold the equator should be north of Deepwood Motte, which is pretty damn far north.
This does assume nature follows similar logic as on Earth. This is an assumption. Hi Edlothiad