"South of the pole" means "all the land that lies beyond the pole in a southerly direction", which as you pointed out, makes no sense, since the south pole is defined as the southern most point.
That's the sensible explanation, but let's have some fun.
So let's define three points on a sphere. Let A be the south pole. Let B and C be two points on the globe that don't lie on A, don't lie on each other and don't line on the same great circle line.
So that gives us a wedge of territory.
If we define south as the latitude, then obviously it doesn't get any more south than point A, which is at -90°. But if we instead define south as a direction, then the argument can be made that if you travel on a line from point B to point A, you're travelling south. If you maintain that direction without changing it, then you're still travelling south.
With that argument "all land south of the south pole" would then cut out a wedge on the opposite side of points B and C.