Direct link to the screen-capture video: https://www.youtube.com/watch?v=mPU6mnM2i-k
TL;DR Hawking only has a single reliable, low-latency binary signal (facial muscle movements), so his interface has been a constantly-moving cursor that he can "click" when it's over the next symbol/command he wishes to select. The innovations here are in the interpretation of those selections: he now has autosuggest for text (designed specifically for him based on the corpus of his works) and shortcuts for filesystem management.
I'm looking forward to seeing when the source code is released, or when a paper is written. Just looking at the data-entry video, for instance, there are interesting parallels between the timing specifications for Hawking's Yes/No dialog and GUI design for dialogs for non-disabled users - in both cases, if there's not enough spacing in between buttons, or orderings are unpredictable, it's much easier for someone to mis-click!