When you see a red anger bubble, open the event first and confirm which Mii this Mii is fighting with.
Then calm the angry Mii before trying to push the relationship forward. Food, gifts, favorite items, travel tickets, or a close Mii can help.
After anger drops, the pair may move into cold war, apology, or make-up events. If the apology fails, comfort the sad Mii first, then wait for another chance.
If you care about the relationship, do not leave a red bubble or cold war unresolved for a long time. Lingering fights can let the relationship keep worsening.