IMPORTANT:
If you were using SkyUI 2.2 or older, you must remove it completely before installing the new version.
Letting NMM handle the updating works fine, too.

Manual Installation
Locate the Data/ folder in your Skyrim installation directory. Typically it's found at Program Files/Steam/steamapps/common/skyrim/Data/.
Extract the contents of the downloaded archive file to your Data/ folder.
In the Skyrim Launcher, select Data Files and enable SkyUI.esp.