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.

In the installer window, select a custom icon theme if you want to, then click Install. If you are prompted to overwrite anything, click Yes to All.

Done!


OR

Manual Installation without NMM

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. If you are asked to overwrite anything, confirm it.

In case you want to use a custom icon theme:
Locate the Data/SkyUI Extras/ folder. In there, pick a theme sub-folder and copy skyui_icons_cat.swf and skyui_icons_inv.swf to Data/Interface/skyui/.

Done!