I too am perplexed as to what the neurons represent, the blog describes the RNN as 19 inputs connected to 7 outputs.
Examining slimevolley_pro.js (Agent.prototype.drawstate) shows only the agent states are depicted onscreen x,y,vx,vy,bx,by,bvx,bvy in the top row and actions forward, jump and backward below.
The code is well documented and builds on convnet.js by Karpathy.
The network is trained by genetic algorithm and self play - so this is a neural net trained with reinforcement learning.
The author's method seems unique and effective.
The blog post comparison of the resultant 'genomes' of network weights seem to show that the space has been searched exhaustively.
The game is played with trained networks but the code contains a training flag - it will be interesting to watch them train with self play.