A made up example to illustrate:
Say 100,000 people from your locale +1 the tweet "she probably deserved to get raped [euphemism for gang-raped and murdered brutally while guys are laughing, recording it on their phones and sharing it with their mates], women need to be reminded of their place, those guys should be let off", and 1,000 people reply with about half of them approving, that's a serious, non-game signal about the real world in your locale.
A signal that tells you something life-threateningly important about the people around you, how they think when they see you, what you should be looking to protect yourself from, and how.
If it was just a game, you could safely ignore the words.
But it isn't just personal safety: Those balance of opinions end up determining how the local police think. Do they investigate, or do they brush it under the carpet? How the local government officials think. What they prioritise, what they even care about. Whether anyone even gets prosecuted, what the law ends up being. Even who ends up in government.
Now, although that's a strong example with details made up (but not unrealistic, sadly), there are countless examples on social media where it is clear that the tweets, likes, comments and other things have a large, profound, physical effect on many people's lives if you trace the effect of the influence of words.
If someone tweets that all left handed users of hacker news who use custom CSS should be banned from the internet, no one gets mad because that tweet doesn't represent some widespread group of humans. It is so outside expectations people will assume it is sarcastic, but even if they were convinced it was real, unless it came from someone in power or showed traction, it would be ignored and none would be mad.
When someone tweets that you doing something you see as natural should result in you being imprisoned under threat of death, and there are entirely communities backing that view and perhaps even a history of people being imprisoned for the same, it gets far more personal.
Arguably, I shouldn't feel stress for things so far beyond my control, but at the same time I have a vested interest in my community and its future (and if so many people feel this way, it certainly _feels_ like a credible threat).
Training is in many ways a body thing as well as mind. E.g. learning to modulate fear and disgust reactions, heart rate, breathe so as to let the adrenalin spike pass (if you want), change muscle posture, etc.
And you should anyway on intellectual grounds, because words' intepretion is often different, at least in degree and shades and layers of meaning, than the intent of their author. For example "$X apologism" is often a misinterpretation of words someone was using to try to reduce the amount of $X in the world (due to the world and people being complicated). Many issues are "no matter what you say, or where you are coming from, someone will think you mean almost the opposite, and sometimes they have a vested interest in thinking that".
But reducing stress doesn't have to mean stop caring.
It can make you more effective at caring.
For anyone with an interest in their community and its future, I'd advocate for deciding which reactions to keep, maybe even develop (e.g. various ways to channel anger constructively, such as politics, community organising, creating new institutions), and training yourself to reduce stress and other intense reactions that you decide are not so helpful (e.g. anxiety from watching the news).
My main point is to encourage people to think of stress and anger as being a mix of constructive and destructive types.
My approach these days is to first try to overcome my anger and despair at knowing how many people seem to think what they do, and when I (eventually!) calm down, ask myself how can I affect the world to change this. I must admit, it is challenging these days because I've come to think of large numbers of my fellow humans as jerks who are happiest when damaging others!