You can use the radio by clicking on it in vehicles. You can change the trigger button on the settings. All local files have to be in the garrysmod/sound/ folder. The URL has to be a relative path to the garrysmod/sound/ folder. Keep in mind that these local files can only be heard by someone who has the file too.

Is it possible to play a sound/music file without them having the file? Otherwise, this will be redundant, as I could just use Wiremod sound output with better results.

Well that radio was firstly meant to be used on online streams/sources which the Wiremod doesn't do (yet). That local playing thing was added after a request. As the API from GMod is locked to be used inside the GMod folder, I can't make it use files outside of it.

Also how is it supposed to play files you don't have? With Magic? That makes no sense at all...

I should ask a more direct question. How do you get it to work with Youtube? I tried that and itshowed the red x near the URL and it wouldn't accept it. Or is there any way I can stream files that I choose?

You should take a look in the console when you try to play/test Youtube URL and it doesn't work. there are some instruction. There are a tooltips on these red crosses, they provide information why the URL isn't valid/not working.