The link in the post above details the method. The fish slowly goes to sleep, and once in a deep unconscious state, stops breathing.
The clove oil is an anesthetic. It numbs and renders the fish unconscious and the fish is in no pain, nor feeling the effects of starving for oxygen: suffocation.
Once asleep, induced gently by just a few drops of the oil in a small container of tank water with the fish in it, then the dose is increased for deeper unconsciousness, and increased to a fatal dose.
It is very sad, I cried when I had to do this to a guppy with severe bloat. I didn't want him to suffer anymore.
I saw a guppy at Petsmart, with a prolapse of his intestines out of his anus, that I know was in terrible pain, and I tried to get the kid working there to euthanise it. His little body was bent from the pain, and he was struggling to swim.
She said. "We don't have clove oil.... I'll just put him in the isolation tank." I asked her to just "get one of the boys that work here to slam him to the concrete floor, to put him out of his misery...", she said, "That's not our policy." Then I tried to buy him to take home and euthanise him. They wouldn't sell him to me. I made it to the parking lot before bursting into tears... I cried all the way home; so sad for that little thing.
If that happend to one of my fish I would immediately euthanise him. There are things that we just can't cure, and some of those things cause a great deal of suffering. I'll do eveything I can to heal, but if I can't, I'll do what needs to be done. Crying as I do it, but nevertheless, I'll do it.