Pull this move is slang and shouldn't be used formally. It just means to do this action/move. Sort of like when you say you "pulled off this move" it means you successfully completed this action/move.
Tired_Design_Gay•
“Pull this move” is referring to the action of giving the look being given in the video. Because the friend says stupid things, they give the person a look we would call a “side eye.”
amazzan•
great example "pull" used like definition 8A (https://www.merriam-webster.com/dictionary/pull): perform, carry out
and the 2nd noun definition of "move" (https://www.merriam-webster.com/dictionary/move), which is more like maneuver
also, it's probably a typo, but it should be "says stupid shit" not "shits." (disclaimer: shit is a curse word)