kinda bullshit

They're not trying to argue that being made to remove the video was a violation of freedom of speech. What they are saying is that this verdict implies that they would have to pre-screen every single video that is uploaded to Youtube in order to make sure it is acceptable before allowing it to go live on the site, which is a ridiculous and unfeasible demand given how popular such sites are. Then there might as well be no site for all the chance there is of getting your uploaded video to be visible within a sensible timeframe.

The problem is that if site hosts are to be considered responsible for the content uploaded to it by their users, their incentive for allowing anyone to upload things is gone. For fear of being sued, that functionality becomes extremely restricted or gone entirely. This is where the curtailment of freedom of speech occurs - by taking away venues for people to speak.