In every job interview I've ever had, when asked to tell something about myself I bring up the massive cake I had for my seventh birthday. You will not believe the size of this thing, or the time and skill that went into sculpting the marzipan princess posed on the top layer in a graceful plié. Almost too beautiful to eat, I had my parents completely fill up the two camera rolls - meant to document the party - with cake pictures. But in the end I'm glad I had a taste because--
Hey, why are your eyes glazing over? This is relevant backstory. You're just like those interviewers who asked me to move on to my employment history!
Backstory is incredibly important to characters for a variety of reasons. Past events shape us in everything we do. Our preferences, the choices we make, the people we love or hate, our bad and good traits, they're all influenced by how we've been hurt at some earlier point in life. A character who doesn't come across as having had a life before the book's first word immediately stands out as fake. But, much like my birthday cake story, that doesn't mean every detail is important.
Like sculpting statues, sharing backstory is a negative art. The master does not add material to the work, but chisels away at the existing layers to show the beauty which has been hiding in the rock all along.
The author could include ten pages of backstory at the start of a chapter to introduce a character. Include a full paragraph of the character reflecting on how much he misses his pre-apocalypse job at KFC, where he'd sneak a chicken wing when he was hungry and the manager didn't look. This isn't engaging writing. It all takes place in the character's head and a potential reader will likely yell 'get on with it' somewhere around page two of the flashback.
Alternatively, have a scene in which the character comes across a stray dog and shoots it, naps a flint, lets out a long and tired sigh, then reluctantly skins his kill. This scene takes place entirely in the present, but the 'long and tired sigh' hints at how life used to be different in the before days.
Does that mean you should never include backstory, ever? No, there very well might be points where you can and should. The 'writing rules' are suggestions at best, but in my own writing I have the 'rule' one of the following two qualifiers must be present for inclusion of backstory to be acceptable.
Is the piece of backstory I am about to include relevant to the scene and the larger story? When a character engages in a fist fight with someone, it's relevant to know the character used to be a professional boxer. When having a tea party with his six year-old daughter, not so much.
Will my story make less sense if I leave out a piece of backstory? Then include it. The fact a character once nearly drowned when she forded a river and is now scared of large bodies of water is an important fact to mention when she decides to take the long way around a river she's currently standing in front of.
Do keep sections of backstory short. If you can't tweet it, delete it. The book you're writing should be about the most important thing to have ever happened in your character's life (or if you intend to write a sequel, the most important thing so far.) If the backstory is more interesting than the actual story, you have a big problem.