Before we dive into extinguishment, perhaps we can back up a little bit and start with a point of agreement.
Westerners do understand and value unselfishness, and that is wonderful. So a conversation about extinguishment might start with unselfishness:
AN4.95:1.1: “Mendicants, these four people are found in the world.
AN4.95:1.2: What four?
AN4.95:1.3: One who practices to benefit neither themselves nor others;
AN4.95:1.4: one who practices to benefit others, but not themselves;
AN4.95:1.5: one who practices to benefit themselves, but not others; and
AN4.95:1.6: one who practices to benefit both themselves and others.
And westerners, having been taught to value self-advancement, will also be pleasantly relieved that the Buddha agrees that there is something better than obsessive self-sacrifice:
AN4.95:2.1: Suppose there was a firebrand for lighting a funeral pyre, burning at both ends, and smeared with dung in the middle. It couldn’t be used as timber either in the village or the wilderness.
AN4.95:2.2: The person who practices to benefit neither themselves nor others is like this, I say.
AN4.95:3.1: The person who practices to benefit others, but not themselves, is better than that.
AN4.95:3.2: The person who practices to benefit themselves, but not others, is better than both of those.
Finally, insightful westerners will be quite relieved that the Buddha advocates for the "win-win" situation as the best of all these:
AN4.95:3.3: But the person who practices to benefit both themselves and others is the foremost, best, chief, highest, and finest of the four.
So what then is this "extinguishment?"
Well, the more we work to benefit ourselves and others, the more we realize just how cumbersome it is to think every single moment, "this is for me and that is for others". And gradually we become dissatisfied with that clumsy way of thinking always about "me and others". Indeed, it gradually becomes simpler and easier to simply think about "best for all".
When we stop feeding the fires of greed, hate and delusion, the fires gradually fade away from lack of fuel. All fires fade this way. All fires fade away from lack of fuel. And in this simple way those three fires of suffering also become extinguished through lack of fuel.
DN16:4.43.4: A skillful person gives up bad things—
DN16:4.43.5: with the end of greed, hate, and delusion, they’re extinguished.”