Some large-scale changes are in progress on the refactor-and-simplify branch. Once these are more stable they will be released as version 0.9.
The more complicated examples have been moved into a separate repository (and eventually a separate Python package on PyPI) from the NEAT library itself.
NEAT (NeuroEvolution of Augmenting Topologies) is a method developed by Kenneth O. Stanley for evolving arbitrary neural networks. This project is a Python implementation of NEAT. It was forked from the excellent project by @MattKallada, and is in the process of being updated to provide more features and a (hopefully) simpler and documented API.
For further information regarding general concepts and theory, please see Selected Publications on Stanley's website.
If you want to try neat-python, please check out the repository, start playing with the examples (examples/xor is a good place to start) and then try creating your own experiment.
The documentation, which is still a work in progress, is available on Read The Docs.