Looks like the GOP SC are trying to allow States to break the First Ammendment and outlaw peaceful protests - by making it possible for States, Police Forces, etc to sue the *organiser* of any protest for any damage done to anyone or any property by anyone attending the protest. This has come from them declining to hear McKesson v Doe - which means the lower court decision on the issue holds. The States that this impacts? Texas, Louisiana and Mississippi
(which is where the 5th circuit has appellate jurisdiction over) for now.
For those not in the know, DeRay McKesson organised a Black Lives Matter in Baton Rouge, after the fatal police shooting of Alton Sterling. During that process, someone (nobody knows who, but it 100% wasn't DeRay, or any other leader/organiser) threw a rock at the police and it one in the face, with the man suffering "injuries to his teeth, jaw, brain and head" (yes, I know, absolutely a little exagerration by the police force there). A previous SC in 1982 had held that an organiser could not be held responsible for the violent actions of a participant; this lower court judgement effectively reverses that (and why the SCs refusal to hear the case is more eggregious given that it basically is giving lower courts free will to reverse previous SC cases).
Why this impacts free protests? Well, if you were wanting to organise a protest, but knew you could be imprisoned, fined, anything if a single participant (even one troublemaker from the "other side") committed an illegal action - why would you ever want to put your hand up to volunteer to organise it (instead of hoping someone else would).