I remember when the cheating scandal came out, there was a post somewhere that basically summed up why Anna can't leave, and it was a really heart-breaking read, because Anna has all the odds stacked against her in this situation. She is a poor, uneducated mother of six young children, she has no real world life skills, no college education and shaky foundations to fall back on. The idea that she could leave and live with her family is a possibility, however, we've seen with J'tyler that the Duggars can provide far more amenities than her family probably could, and so if there was a battle for custody, I'm sure Meech wouldn't turn down the opportunity to have six more kids in the house to fill up those empty beds. I imagine Anna was lied to.
Put yourself in her shoes. She grew up poor, in a trailer park, no job prospects given her shameful status as a woman, no way of supporting herself. And then, along comes Josh, a semi-famous, fairly wealthy man who can lift her out of poverty and provide for her, and maybe her kids wouldn't have to grow up as she did, at least financially. She finds out about his past, but the allegations are most likely presented to her in a very minimized and diluted way, she's assured its all in the past, and so she accepts it and moves on. Anna marries Josh, and sure things aren't perfect, but she's out of the trailer park, she's in a nice house with a man who at least publicly, pretends to love her, and she can finally fulfill her destiny of being a mother, the only things she's told she's good for. She insecure, Josh is 'Josh Duggar' and she may feel like she's reaching, but she's just so happy to have him that she shoves all that insecurity way down deep and gets on with her life. Life goes on, they move to DC, she's living a cosmopolitan (at least for her) life, Josh is a 'politician' for their awful beliefs, and she's a politicians wife, at home looking after her kids, who she loves completely. Sure, they may have less money than she thought (for reasons she can't bring herself to admit) and Josh starts spending more time on the computer, and staying later for work, but hey, she's just a helpmeet, and she'll be there when he gets home.
Then the scandal hits, and she finds out everything, and all that insecurity comes rearing back, and her world is shaken to the ground. Everything she thought she knew about Josh is turned on its head. Anna realises how little she knows off the real world, or maybe she always knew and didn't want to confront it, the stuff he does on the computer she never understood and so didn't bother with it, the places he claimed to go sounded true to her, so she went with it. She places all the shame on herself, because that's what she's been taught to do, she had one job, to be a good wife and mother, and she's failed at it. Anna put all her eggs in a Josh sized basket, and it didn't work. They have to give up everything, run back to Arkansas in shame, and live in a windowless room with the man who betrayed her in every way he could, but maybe the lack of windows are a good, thing, because no one on the outside can look in and see how miserable she is. She has six kids by now, seen her sisters-in-law get married off to better men (everything is relative). She sees Jill, with her husband who has an actual degree, going to therapy, healing and loving her sons, and yet she remains in her cage. If the scandal showed her anything, its how little she knows about the world, so she doesn't leave, she sticks to the only thing she has ever ascribed to and hopes it will bear fruit in the future. All she has is her kids, and she's going to hold onto them with everything she's got, and if that means not leaving Josh, then she'll do it.