The tvOS 12.1.1 public beta can be obtained by going to the Settings app on the Apple TV and navigating to the Software Updates section under "System." "Get Public Beta Updates" will need to be toggled on, and once it is, the Apple TV will download the beta software.
What's new in tvOS 12.1.1 is a mystery because Apple does not provide detailed release information for tvOS updates. The new software most likely focuses on bugs that were not able to be addressed in the tvOS 12.1 update.
Nothing new was discovered in the tvOS 12.1.1 beta provided to developers, which is not a surprise as Apple's tvOS updates have historically been minor in scale.