Displays in mobile devices like smartphones are getting to be higher quality at a very fast pace. It was about four years ago when we got displays with 720p resolutions, resulting in a pixel density of more than 300 ppi, with a Full HD resolution in a 5.5-inch screen it results in more than 400 ppi and there are even QHD displays which sport a higher resolution than most TV sets at 1440 x 2560, resulting in more than a 500 ppi density. This basically means that pixels are no longer visible unless you look at the screen really close and you really want to see those individual cells that compose images.
The Android operating system was optimized with the introduction of Android 4.1 Jelly Bean with the so-called Project Butter, which made the animations a lot more fluid and smooth by using some processes and running the whole interface at 60 fps, so the display technology has been able to show this kind of frame rate for a while. YouTube has been supporting videos at 60 fps for over 8 months but it was limited to the browser video player that uses HTML5, since May, it even supports this kind of frame rate for live streams.
Now, the YouTube app for mobile devices is supporting the 60 fps frame rate, and there's no need to update the app as Google is updating the content from the server. Most videos are displayed at 24 fps or 30 fps as It comes from movies or TV shows, but video games use 60 fps, so videos of game plays or tutorials can get that benefit on Android and iOS devices. Unfortunately, those recording their live streams or wanting to watch them at this frame rate will have to wait a little longer, as it wasn't said when will they be supported. After a video recorded in this frame rate has been selected, users can adjust the resolution to view it the way it was intended, it is clearly marked as 720p60 or 1080p60 so HD doesn't support the lower frame rates anymore. This move could also make an impact in VR headsets as videos would appear smoother.