You use a neuronal network to decide at each step to flap the birds wings or not. That is the only output. The input is only the birds height (y-position) and the height of the next hole. (https://en.wikipedia.org/wiki/Feedforward_neural_network)

The question now is how to find the correct weights for the net.

That net is not trained in a traditional supervised way like with gradient descent (which would be more complicated). Instead it uses a genetic algorithms to find new weights for neuronal networks of future generations. (https://en.wikipedia.org/wiki/Genetic_algorithm)

And that's basically it in this case.

You can see some of the older NeuroEvolution papers here: http://nn.cs.utexas.edu/project-list.php

I believe Ken Stanley has gone to become a professor at UCF to continue his research: http://www.cs.ucf.edu/~kstanley/

Otoro explains neuroevolution and demonstrates a Silme Volleyball opponent evolving