It appears to use Recurrent NNs (RNNs) that have a 'Long Short-Term Memory' (LTSM) architecture.
Here's a summary of the development process that the author, Ross Goodwin, went through to create it.
It seems to me (and is also observed in the above link) that the output is rather poor - simply comparable to what one might expect from Markov chains, a technique that is over 100 years old.
I haven't dug deeply into the technique, so I could be misktaken, but perhaps one of the reasons that it's so bad is that (as far as I can see), the model-building process is essentially lexical - i.e. it is linking together tokens (words) without any more informed language model to guide it. In particular, the generated output doesn't seem to know anything about the functional roles played by objects (chairs are supporting objects, used by humans for sitting on etc), which is something that might be fairly readily incorporated.