How does the App "Battery Repair" work? I just found this App (Battery Repair) on the Google Play Store, my question is how can it make my battery last longer. What optimazions are they doing? Has anyone tested this tool, does it work?
No app can "optimize" or "maintain" the battery in any way as it can't get the necessary access to the hardware -- probably not even if it required root. This app explicitly advertises that it does not. Anyway, the manufacturer's engineers surely have already implemented all the necessary optimizations. There is no way that the app's developers knows anything useful about every phone's charging circuitry and battery that the engineers don't.

In short, there is no way this app does anything useful at all. (It might even be malware, but I don't get that impression in this case).

(Note: the following is probably not true)

The only thing an app can do (on a rooted phone) is to recalibrate the battery. If the battery is miscalibrated the OS might think that it is as good as empty and force a shutdown although there is still a useful amount of charge left. One app that does this is the aptly named Battery Calibration.