Very simply, the brain is always a flurry of biochemical and bioelectric signaling, even when you're thinking of "nothing". When a person is trying to do something, they focus on the subject or activity, pathways in the brain light up and neural connections strengthened. When a person stops consciously focusing, the brain doesn't "shut down the system" so to speak. Activity triggered by the active work continues to happen, but in a less organized way. That activity is can end up making connections to ideas the conscious thought process had not even remembered existed.
That's how solving problems happens after "giving up", and this is well-documented in cognitive studies. It's called the "default mode network"[1] and it's possible to learn to activate it.
As for your example "Give up trying to cure your alcoholic and your partner decides to go to AA on their own", that's an entirely different mechanism, involving human social relations and self-interest. I don't have a good explanation, but have you ever been in a situation where someone told you to do something that you knew was good for it, but you resisted doing it? Maybe you justified your refusal as being skeptical, or just didn't like being told what to do.