The biggest appeal of post-apocalyptic stories is that they show us what might be, and serve as a cautionary tale. Stockpiling nuclear weapons and spreading biological infections that turn people into zombies is a bad thing, but makes for an interesting story especially if it's presented in a realistic way. Post-apocalyptic tales are also very entertaining because they tend to focus on individuals who overcome their barren, solitary existance and begin anew, like pioneers or early settlers. Sometimes the character even manages to save the entire world through his or her actions, showing that a single person can make a difference. Despite the setting, the stories can be hopeful, and show that no matter what happens, nothing can really squash the human spirit and our stubborn persistance.
Another thing about the end of the days is that it involves the purging of mainstream culture and starts fresh. The old is wiped away, and a new order is put in place. Mostly, the stories are a return to nomadic lifestyles with no massive cultural centers, and government is abolished. The anarchy that replaces the old system provides much more personal freedom, and a person becomes committed to the family and the self, over the entire society. Theres something appealing about getting away from it all, moving from the humdrum life in the city for a quiet, personal life among the plains. I think that's why most post-apocalyptic scenarios depict a vast, sprawling wasteland of land or water, rich with opportunity and untamed once more. Despite looking like a horrible nightmare world of grays and browns, green water and crumbling buildings, the stories almost always take place after the radiation has settled and at the point where people are starting to reclaim the world for their own. Part of the fantasy for me when I play a game in the Fallout series is the exploration and personal achievement that comes with taming a new world. Cultural centers are at most a collection of shanty style huts in a crater, and at times just two homes made out of spare parts and old world know-how under a ruined bridge.
The fun is in surviving and making a new world, and by having the old world completely obliterate itself shows just how faulty and full of problems it was. The allure of a new world is that it's made by the few who could survive, and who possess the knowledge of the past and know how it came to disaster, and who know how to avoid it. Any problems with the world will be washed away in atomic fire, a real cleansing. You could say that mainstream culture needs a reboot, that it's inevitable that we blow ourselves to bits, and that we should just as well look forward to it. Stop worrying and love the bomb.
Another thing that survival stories of this kind do is prepare people mentally for a nightmarish scenario, and show them what they can expect. The best part of realistic science fiction of the kind Asimov employed is that it takes a very focused look at our current society and extrapolates on it, predicting a future if things stay on course. The resulting future can be analyzed and thought upon, and affect how we deal with it now. Genetic modification, robot slaves, time machines, dinosaur themeparks: how they fit into our world and the changes made can be examined, and if these things sound harmless at first, they quickly appear to have deadly side-effects that are best avoided. If one of these things is unavoidable, talking about it through story can at least prepare someone for the future. A recent video I watched stressed the importance of violent videogames in a child's education, to teach them how to survive in the future. While humourous, it may not have been totally off the mark, since it does get you thinking about how you would adapt later on.
Primarily the stories are about human endurance, our strength at adapting, and getting away from the culture in order to rebuild it from the ground up. The backdrop of a nuclear wasteland just serves as a means of telling a story of retaming the world for a better future, getting the individual more attuned to nature, and appreciating nature for all it's given us. In exchange for a world made by many, you would get a more responsible world made by the one.