First off, you'll need to subscribe to the addon, so if you haven't done that head over here.

Right once thats done and dusted you'll need to download the loading screen replacement from:
http://xenthio.github.io/loadscreen.zip
(Garry's Mod doesn't allow me to include the html files inside the actual addon, so this is sadly necessary.)

In another window, open your gmod installation folder and open the garrysmod folder (by default that is C:\Program Files (x86)\Steam\steamapps\common\GarrysMod\garrysmod)
It should look a little something like this:

Open the zip and proceed to drag the HTML folder from the zip into the garrysmod folder

If you are asked to replace files, you must replace them.

If followed correctly, you should now be able to boot up the game and enjoy some better than default level transitions.