That’s not how you’d do that.
If you want to grab clips from youtube and then edit them for home movie gag reel fun, get the free application/ web app called clipconvertercc.
You drop the URL of any youtube video into clip converter via copy/paste, then you can extract and download a version in MP3 audio, or various video formats, which you can then edit with. You can capture and download entire clips or designate by run time the start and end of the clip from a longer section.
Once you have the clips captured, you can edit them in FCP7 or FCPX. FCPX would be easier, since it will handle any clip format you throw at it, and the older FCP7, long since EOL and discontinued, may require the clips to first be converted to a format FCP7 prefers. Apple Compressor can handle those conversions, (and if you have FCP7 you likely already have Compressor) or use a freeware app called MPEG Streamclip to convert weird or newer-format files into pro res files that FCP7 understands.
One other way to capture youtube type clips would be to play them large on your desktop, and using the free screen recording feature within Quicktime Player. Go to Quicktime Player, File>New screen Recording, drag a window around the frame of the youtube player screen, hit the record button, stop when done. Save the recording, import it into Final Cut, then in Final Cut, use cropping tools to get rid of any unwanted edges.