Part of my 2008 honda fit compressor is not working as shown in video. Not sure what the next step is or which video to watch to help. Wondering if I can fix this by myself.

 

That center part is called the compressor clutch. The compressor has an electromagnet that makes that center part pull in and lock to the spinning pulley when electricity is applied. There may be a problem with your A/C system, or with the clutch. There's a 2009-2012 factory service manual floating around here some place - the compressor looks the same as yours.