In any group, people will have problems.
If you're the authority within that group, you're the first stop for problems.
Don't deny the problem exists. Don't apologise (not yet). Don't claim you're doing your best. Don't claim you're covered by guidelines. None of these are helpful.
Just focus on clarifying the problem.
Repeat the problem back to them. Literally, tell them the problem they just described in your own words.
"to make sure I understood this correctly, the problem is ....."
Then ask further questions to clarify the problem. Ask what they want you to do about the problem. Gather as much information as you can. Then go see if you can resolve it.
The more you try to understand the problem, the less the member feels you're part of the problem.
You can't diffuse the anger until the other participant feels you've really understood them. Simply by making a genuine effort to understand them, they feel you're on their side.
If I knew this tactic 10 years ago, I would've saved myself plenty of anguish.