The Mari/o video of SethBling was an excellent introduction to how it works back then. [4]
As NEAT is an evolutionary training algorithm, behaviors are represented with genomes (or phenotypes), and the performance differences of each agent is evaluated with a CPPN (compositional pattern producing network).
The strength of an algorithm like this is not sensorics (like an LSTM, RNN, FFNN etc) but the combinational aspects of "which sensor to trust" as a meta learning concept.
[1] https://www.cs.ucf.edu/~kstanley/neat.html
[2] http://eplex.cs.ucf.edu/hyperNEATpage/