The basic jist of it

To subscribe to a mod, you simply need to click on the "Subscribe" button.

On the right side, you will see the "Required Items". These are EXTREMELY IMPORTANT, as a mod will not run unless you are subscribed to all the required items. This sounds obvious, but 8 times out of 10 that's the cause of a mod not working properly. (1 time it's because the mod itself is broken and the other is because Steam just hasn't downloaded it properly)

If you subscribe only to the main mod and don't have the required items you will get this window here, telling you that will also need the following items
But no button to subscribe to all of them, thanks Valve

You can actually click on these items and Steam will take you to their own pages
Though it's a bit clunky to navigate, again, thanks Valve

If you have all the required items, a green tick will appear next to each of them. Now it's ok to subscribe to the main mod.
As the notification says, sometimes you'll need to restart Steam for it to download the mod

While not always required, it's always good to read the description of a mod, since it may contain important information. Be considerate of the author and think twice before asking a question which has the answer in the description.