When you download a mod in the Mod Browser, it won't install it immediately. But when you go to the page where you select which mods to use, it'll have a giant "Install Mods" button. Press that, and it installs all mods you've downloaded.
Then just select the mods you want, and you're ready to go. There's a circle next to each mod's name; click it, it'll turn green, and the next game you start will include those mods.
Note, though, that many (most) of the mods on the in-game mod browser are horribly out-of-date and just won't work. The game has fundamentally changed a few times in the year since it was released, and many mod creators have not updated their mods to work with the latest patches. A message board like this is the best place to keep track of which mods are active and which aren't; if the mod has no discussion thread here, or the thread in question has been dormant for more than a month or so, then chances are the mod is no longer supported.
Also, many mods are inherently incompatible. The larger the mod, the more likely this is to come up. No two mods can adjust the same UI element, for instance. So don't just download every single thing that sounds good and try to turn them all on at once; it probably just won't work.
