Beta Tester here.
So the article pretty much sums up how this aims to save you money: by making you drive slower.
In short, if you tend to drive in the passing lane for the majority of your commute, like I do (or used to

), the little device's benefits become immediately apparent. I'm getting a full quarter tank more than I was before.
The software works very well- or it used to up until two updates ago at the start of August that has rendered my app useless. As for the device itself, it works regardless of whether you're connected or not, and the beeps are pretty unobtrusive, so I've still been able to benefit from the fuel savings. The parking location feature is nice as well.
My largest gripe with the Automatic is that it requires Location Services to be turned on at all times. I hope they find a workaround for this (they do explain the reasoning for this on their forums), because it does cause a significant battery drain even while you are not driving. The cure for this is to simply disable the Location Services for Automatic when not driving in the Settings -> Privacy menu.
All in all, I'm glad I have this device. It's not for everyone, but for me not having to fill up twice a week two times has already caused the Automatic to pay for itself. Very satisfied.