What is 'won't' used for?

Solution 1:

The English verb will is related to the noun will, which is one's capacity to want or wish to do something. Originally, the verb had a meaning similar to want/wish, as it still does in Dutch: the verb willen means "to want".

Then at some point its meaning began to partially shift from desirability to probability: he will do it means "it is highly probable for him to do it in the future" rather than "he wants to do it now". It is common for probability and desirability to be somewhat fluid and shift around in all European languages I know. See this question, whose answers contain a more thorough explanation of the shift.

So the verb will changed from wanting to future probability, but not quite: in certain uses, it still (partially) has this old meaning of desirability. The type of sentence you quoted is a good example:

He won't come out.

What this really and originally means is "he doesn't want to come out", the old meaning of the verb. Perhaps the sense of wish or desire is not entirely clear any more to most modern speakers, but it is the reason behind this construction. The fact that we find it odd when we think about it is testimony to that.

Solution 2:

The "will not" or "won't" sometimes implies it's the choice (will) of the subject. "He isn't" does not. Nevertheless, "He won't" is idiomatic for the same thing, at least in modern usage.