There are a lot of sources of information about the Quiverfull movement online. Here's a sample of what they believe:
"From the moment a Duggar in possession of a vagina is born or married into the family, she falls under the control of Christian Patriarchy. The wife is forever under the authority of her husband, and a daughter is completely subservient to her father until the day he finds a suitable husband to take up the role. The women of Quiverfull are never supposed to disobey their male authority figure, ever, even if it’s something they find immoral. Because they think God would rather you kill a guy because your husband says so than to disobey the orders of a man.
Women aren’t supposed to work outside the home, and if the father can’t find anyone for his daughter to court and marry, they have to live at home to take care of the other children. (Sorry, Jana Duggar.) College? What’s that? There’s no room for individuality for men or women when you’re immediately dictated into a role based on the kind of genitals you have.
...women are obligated to give it up to their husband whenever and wherever they want it. Michelle Duggar herself relegated this advice to her daughters Jessa and Jill, on Today.
“In your marriage there will be times you’re going to be very exhausted. Your hubby comes home after a hard day’s work, you get the baby to bed, and he is going to be looking forward to that time with you. Be available. Anyone can fix him lunch, but only one person can meet that physical need of love that he has, and you always need to be available when he calls.”
So, here's the thing: These children are "home-schooled;" i.e. are isolated from society and are then fed a bunch of vicious baloney about the world. Part of that is the notion that women are nothing but breeding machines, and that it is their responsibility to have sex with male authority figures any time the males demand it. When Josh, the oldest male in the family, learns by example (being systematically denied experience of any more healthy adult behavior) that this is the ordained relationship between men and women, and that women only exist to serve the needs of men, why would it be one bit strange- in fact why would it not be predictable- that he should follow the example of his father, and decide that the women around him exist only for his pleasure? Like the Catholic priesthood, here's another example of Christianity being, in fact, a breeding ground for the most vile of perversions.
I just want to add, in finishing, that we are brought up to grant extra respect to beliefs that stem from religion. In today's world, the exact opposite is true. We have learned through painful experience that, when someone justifies their conduct on the basis of religion, the proper response is to assume that they are up to something they can't justify in any real way, and are just using a dodge to persist in behavior that we all know is repugnant.