The first part of the video is with the 1.1, or 1.11 version of the software (I forgot to write it down). The second half is with version 2.20.
Version 1.20.1, released at the end of 2017, added "a function to correct rotational shake when shooting video.(When transferring to a smartphone with zenith correction turned on)" and "Subject shake when shooting video has been reduced." Version 2.20, released May 2018, added "When playing video with the remote playback plug-in, the video will now be displayed after rotational shake correction."
The stabilization isn't quite as smooth as the like-it's-on-rails Rylo, but it's definitely an improvement, smoothing out the bounces and shakes enough to make walking around with the Theta V far less annoying to watch.
Since this was a firmware update, not a feature added in the software, there's no convenient way to show the same video with the correction on and off like what's possible with some other cameras.