I understand your frustration as I too was looking for what series of hypotheses her journey of discovery took her through. Alas, this was a (good) human interest piece so that was not the focus.
Assuming this device works, it is fun to think about why this might be. All I could extract from the article is that it sends vibrations and fails for some, requires randomness in others and works at a specific pattern for Lawton.
I am not an expert but what I do know is that Parkinson's can be viewed as a breakdown in the proper transmission of feedback from erroneous predictions. In planning movements, the brain is constantly making and correcting its predictions, sufficient damage to part of this circuit would result in improper gain modulation and ineffective dampening to prevent overshoots. Improper cost predictions might even affect motivation to perform a movement in the first place.
It's not clear how a pattern of vibration might lead to a reduction in symptoms. My best guess is that it's related to stochastic resonance. The noise or pattern from the vibrations, and that the population of neurons responding to the vibrations are different enough might be such that, whatever weak signal there is carrying the proper feedback is now more readily decoded. Whether or not this device works will likely be highly variable, dependent on the state of degradation and particulars of the erroneous feedback control.