The tvOS 12.1 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.
We don't yet know what fixes and changes the tvOS 12.1 update might bring, but it likely focuses on bug fixes for issues that have been discovered since the release of tvOS 12.
Apple's tvOS updates have historically been minor in scale and Apple does not often provide us with detailed notes outlining what's new. No obvious outward-facing changes were found in the first developer beta.