How to explain to someone with no knowledge about Buddhism what is meant by "extinguishing" the self?

It's foremost good to get really adequate to the teachings by oneself, as, for example, the Sublime Buddha taught nobody the extinguishing the self, and teachings are gradually taught as well, good householder.

So for now it would be good to, if by oneself allready that firm there, to speak in praise of virtue, of reasonable goodwills benefit and the blessing of generosity, as well as what/who would be worthy of gifts and why.

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.”

Here is an analogy suitable for a layperson.

Imagine you are dreaming. In the dream, you don't recognize that your 'real body' is lying on the bed, you genuinely think that you are the dream character. A tiger chases you in the dream. You think the tiger that is chasing you is real. You run from it, it makes you suffer. You ask yourself "Why does life have to be this hard? How to get the tiger to stop chasing me?" but it doesn't stop.

Then you wake up & realize that the dream character and tiger and events that transpired in the dream are all illusory. There wasn't really a tiger. It certainly felt like there was, but now you realize it isn't substantial. Suddenly the tiger isn't so scary anymore. You realize whatever happens in the dream can't have an effect on you, once you recognize the nature of the dream.

Extinction of the self doesn't mean extinction of feelings, thoughts, emotions, situations, body, mind etc. It just means realizing their true nature. In the dream, you can experience emotions, feel things, experience happenings. But they don't harm you if you wake up. Similarly, when you realize what you are beyond this body, beyond this mind, suffering ceases. When you 'extinguish the self', you still have a personality, feelings, connections and various experiences. But they don't harm you because you understand their nature.

Let's say you know someone who is strongly addicted to recreational drugs (like cocaine or heroine) until he cannot live a normal and productive life anymore. He looks disheveled and constantly seeking a state of high, otherwise he would experience withdrawal symptoms.

You approach him and say, "would you believe me, if I told you that it's possible to live a normal, healthy, happy and productive life, without drugs?"

The drug addict may reply, "What kind of life is that? It would be like living in unfeeling lethargy". The drug addict thinks that the lack of feeling high by drugs must then be some kind of unfeeling lethargy.

This is similar to the situation that you have described, with respect to the ordinary person thinking about those who have become liberated from samsara.

Another approach would be to peel the onion. The Heart Sutra does this in much greater detail.

I am not the body. I am not the senses.

I am not the mind. I am not the thoughts.

I am not the intellect. I am not the emotions.

What am I? Who am I?

When all the layers are gone, all that remains is a sense of being that is entirely impersonal.

I am not there.

When I look inside, I see that I am no-thing. That is wisdom.

When I look outside, I see that I am every thing. That is love.

Between these two, life flows.

-Nisargadatta Maharaj