No "Options"
 

I can't seem to download any files. When I double click on them, nothing happens.
On the FAQ page it says that I should go to Tools>Options...
Problem: I've got no "Options". Under "Tools" there is only "Statistics".

Why is that?

P.S. I'm using mac and the basic version.

Sound like a license issue, see the options here License requirement (click on link)

On a mac your prefs/options are found under the LimeWire name next to the apple icon at top left of screen. Keyboard shortcut is ⌘ ,

BTW the preferences/options of all programs on OSX are found in the same location. And all use the same keyboard shortcut.
However if you're using OS 9 or 8 then you will find LW's options in same place as windows.