IIRC it was a special condition of cosmic rays, higher than normal or combined with the equipment or something like that. Needless to say, I don't think it needs to be "explained" much more than a handwaving. It's rather small and insignificant part of the story, overall.
I sort of agree: Any explanation will be handwaving and it's a small detail. But it's still significant in allowing us to suspend disbelief. A genuinely good explanation would be better writing than a merely sciencey-sounding one, but truly rationalizing this stuff is beyond any human. But whatever fictional science is used needs to be well-written fictional science.
I don't like the Nolan movies. Chris Nolan wouldn't dream of sticking his hand into a working microwave but can't see how silly his microwave weapon was. But we have to "see" microwaves both working and not working at the same moment. And he did stuff like that while confusing drab or confused or merely cynical with realism. Feh.In Nolan's more "realistic" Batman movies....
How did the Fantastic Four's exposure to cosmic radiation impact them and not others who went into orbit before them?
Well, the early comics clearly established that cosmic rays would give other people super-powers, and their pet apes too. The Red Ghost. Perhaps I'm mistaken, relying on memory, but unlike you I don't think the early comics ever answered the question, despite man's continued march into space.
I don't think it's excessively demanding to expect a new movie adaptation to try to clean this origin up a little.