The premise is simple: Nuclear Armageddon happened and humans brought it upon themselves. Now humanity has to survive in a new, harsh environment, but without the Old World capabilities.
The underlying theme is more interesting: it's about the Old World legacy, namely it's technological marvels. Vault 13 is the best place to live in the entire Fallout's California due to being constructed before the war with all of its technological achievements, but at the same time it's very fragile: malfunction of a rather simple water chip is enough to doom it, and there are no ways in replicating this component. Then it's outside menace: Vault has little protection aside from it being hidden and unknown to anyone, so the simple fact of revealing its location can doom it.
The Old World is gone, and while its heritage is still crucial to survival, people should learn to live without. Notice how Fallout 1 shows the dichotomy of old and new life: Vault 13 as the example of old ways, comfortable, but with little perspectives in the future, slowly stagnating; Shady Sands as the new way, with almost none of pre-war technology and lots of hazards, but having opportunities to grow and attract new people. Vault 15 shows what can happen with Vault 13 in the future, simultaneously showing that this would not be the end: life in the Wasteland is quite possible.
Then it's the dangers of the Old World technology: supermutants, despite being a new threat, completely dependent on Old World heritage. They were created with pre-war FEV, they use pre-war weapons and equipment, they use pre-war facilities as bases and such. It took only one man to use the existing capabilities and threat the life with extinction one more time. As if Nuclear Holocaust wan't enough, Old World reached from the grave to extinguish the life once more.
Brotherhood plays an important role here too. They hoard technology, preventing it from being used in the wrong hands. The events of the game force them to abandon their reclusive politic and get involved into the outside affairs or face destruction.
So, Old World is gone. It's futile to try to bring it back. Use its legacy while it still exist, but do it cautiously and learn to do without. Society should be rebuilt and isolationist policies will not work.
Fallout 2 partially follows this narrative. VD went North and full Luddite, probably after realizing the dangers of technology (and it's ironic that his descendants realized that they can't do without). Enclave, being an ultimate threat, is one more relic of the Old World. Vault City is a good example of "use it, but learn to do without" mentioned above, they opened up and avoided stagnation. And, of course, we're shown the ultimate fate of Vault 13: it stood untouched right before Enclave came in and took everyone warm.
