The play brain is the part of the mind that only sees a game in terms of levers and buttons, actions and reactions, and its only interest is in self improvement. It sees the frame of the game rather than the richness of its detail, and forms a purely literal understanding of it. It is object-driven rather than metaphorical, deductive rather than dramatic, and it has little interest in anything that isn’t speaking to it on its level.
It realises that all first person shooter enemies are just cylindrical objects that take three shots to kill. It sees through the pomp and ceremony of Shogun Total War 2 and grasps that all the troops on the battlefield are just dots that move on command. It figures out that all crops in your virtual farm are just clocks that will spit out coins at the end of a cycle. It is also the part of the player that grows bored, is relatively sociopathic, reacts negatively to unfairness and derives excitement from the learning.
While the play brain sounds somewhat like the left brain (and may well prove to be the case), the distinction is made to separate concept from actual discussions of biology.