If you're unclear what Live Photos even are, they're like regular photos, except when you hit the shutter button the camera captures a still photo, plus about 1.5 seconds of video before and after it. To view it, you press down on the photo and the whole scene comes to life. They're pretty damn neat, except that Live Photos you take in motion usually come out pretty jerky. Motion Stills solves that issue – even better, it turns them into a looping GIF.
The first time you open the app, it'll pull together a library of all the Live Photos you have on your phone (so even if you don't have a 6s, but have been sent Live Photos from people who do, they'll show up). As you scroll through, you'll notice the app has automatically turned them into GIFs, and smoothed 'em out by running them individually through special video stabilization software. Suddenly, that Live Photo you took while jogging looks like it was shot on a Steadicam. If you want to see just how much smoother they are, tap on any and hit the semicircle icon in the bottom-left corner to remove the corrective stabilization effect.
If you want to show one off, tap the share button and share the GIF via text/email/Twitter/carrier pigeon -- finally making it easy to show off the full effect of your Live Photos with iOS-less pals.