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.

Automatic Download with NMM
Click the Download with manager button on top of the file.
SkyUI will appear in in NMM's Mods list once it's downloaded. Double-click the SkyUI entry to Activate it.


OR

Manual Download with NMM
Start NMM and click on Mods.
In the left icon bar, click on Add Mod From File and select the downloaded archive file.
SkyUI will now appear in the list. Double-click to Activate it.


OR

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.