@moonbolt
From your description I mostly get that you should probably (some other guesses below; this is all guesses without a back and forth of questions) try internalizing your understanding of other's anger. You knew they were angry, you knew it wasn't at you, you knew they wouldn't lash out about this at you. If that last one isn't true, then that is another good point of intervention -- either they might, and then they definitely need to work on that and you are justified bringing up you feel unsafe in such situations, or they don't and you have to somehow convince yourself of that (perhaps by getting to know them better?). Anyway, if those three are true, then the anger has no reason to feel uncomfortable to you, so it shouldn't if you have them internalized. Unfortunately, I'm not sure how to achieve that in general -- the thing that works for me is reminding myself of such a reasoning during relevant situations and sometimes also when imagining them, but I don't know what will work for you.

Another possibility is that you are personally fine, but feel for the innocent being hurt in the situation (even if they are hurting themselves). I see two very different approaches to dealing with this, although both might be applicable at the same time. One is to accept that people have quite robust authority over themselves, and in particular can direct anger at themselves for whatever reasons they see fit (although the main reasonable one is self-modification). While this is likely true, it has obvious limitations, self-harm exists and wanting to stop it is natural and good. Which leads to the second approach -- if the anger was self-harm, bringing up that you think they are being too harsh on themselves is likely reasonable and, if done delicately, shouldn't hurt them by itself.

I'm not sure any of this advice is useful, but I hope at least the general perspectives for looking at the problem are.