If you're setting your story after an apocalypse, readers are likely to be curious what happened. If your setting is several centuries post-apocalypse, it's not unreasonable that nobody would know, and it is less relevant to the ongoing story anyway. If, however your setting is only several decades after the apocalypse, some of your characters would have been born before/during the apocalyptic events, the whole thing would still be within living memory. The effects of the event would still be felt, its effects would still be present. (For example, if it were zombies, people would still maintain anti-zombie precautions.)
Do you necessarily need to satisfy the readers' curiosity and answer all their questions? Not necessarily - it depends on your story. But you need to know what the nature of the apocalyptic event was, because of the consequences of the event would be different depending on the event, and you need to be able to write that. If you know the event, and its particular consequences are present, that might be enough of a hint for your readers to figure out what happened - you don't need to tell them directly.
Now the backstory of the characters - again, as @Cyn points out, you need to know what their backstory is, because that backstory would shape who they are now. The apocalypse would be part of that backstory, whether they were born before, during, or after. How much of it you tell? As much as the story requires.
Another important note: you don't need to tell everything straight away. Some things can start out as a mystery, to be gradually revealed over the course of the story. If you've watched Firefly, you might remember how the backstories of the different crew-members are revealed differently: Mal and Zoe's story we know straight away, it's key to understanding them. We know Simon's story from the first episode onwards, but what's been done to River is an ongoing mystery that we learn bits and pieces of as the story progresses. We get hints that Shepherd Book and Inara have backstories, we are curious about them, but some muttonheads cancelled the show, so we never find out. And as for Wash, Jayne and Kaylee, for all we know - there's not much to tell. You can treat the backstories of you characters the same way: one character's story is interesting, so you reveal it when it's interesting and relevant, all together or in bits and pieces. Another character doesn't have much of a backstory, so you don't tell it.