I don't know how to do that - except by refering to personal experience, history, law, and science - and I don't believe anyone else does either. That's where you and I part company.
there are individuals who believe the answer is, "Best for me, according to my pleasures." These people exist in all cultures.
That's right. And they cause trouble is all cultures...because their short-sighted selfishness undermines community, law, family, continuity; things which are valued in all cultures.
Are they wrong? If so, how do you know? If not, how can you claim to have explained morality in any sense, with or without the supernatural?
I can act in accord with my conscience. A culture act in accord with its norms. That's the best that can be done. Here again you and I part company.
It seems we do part company. But I do have one request. If you cannot believe, let alone defend based on your worldview, that some things are simply right and others wrong, for all individuals and cultures, at least stop claiming you can explain morality without the supernatural.