Google Photos has finally added video stabilization in v2.13 after announcing that it would be coming to the app but not giving users a date to work with. To get to this feature you'll first need to have a video stored on your device, and if you don't have a video that's located in the local storage then you can simply record one or download it from your Google Photos online storage. Once the video is stored locally you can open it up and hit the edit button where you'll find the ability to stabilize the video as it will have its own dedicated button sitting below the video frames.
It's worth noting that going into the video to edit it even if it isn't stored locally will download the video for you so there also isn't a need to download it to your device first, but you can. Once you hit the stabilize button you simply have to wait for it to finish and the timing here will depend on how long the video itself is and at what resolution you recorded it at, as well as a couple of other factors. Overall it shouldn't take too long and in many cases the length of time to stabilize a video may be almost equal to the video length itself.
Even though it doesn't take long to stabilize a video for the most part, Google has implemented a progress bar that will show you how far along the process is although the bar doesn't show you a percentage. After the process is complete you can compare the stabilized video to the video that isn't stabilized if you care to see the differences between them and you can do this by tapping on the stabilize button again, which essentially toggles it on and off. In addition to the new stabilization feature, Google may be looking to add some sort of Google+ integration back into Google Photos as some strings of code that reference Google+ were found in the code of the latest Google Photos APK. Having said that, this doesn't mean Google is working on integrating the two services again. If you want to grab the latest version of Google Photos that has video stabilization included, you can grab the APK from the button below, although the update should already be live on the Play Store.