What do they mean and what purpose do they serve?
They're tags that are used on MathOverflow. The reason they were adopted there is that professional mathematicians are already familiar with the two letter arXiv codes for various areas of mathematics.
I don't think it makes sense to have these tags on math.SE. That is, I think [pr.probability] should just be changed to [probability]. A moderator can do this easily, so there's no need to go on a retagging spree.
This is a carry over from Math Overflow.
However, it seems as though it is an impossible-to-enforce convention here.