You can see more of the Bates family on their website.
Having so many kids is a Quiverfull thing. On the face of it, they use the Bible (fairly superficially and uncontextually) to justify using no contraception. Really, though, it's an attempt to breed as many white, straight Republican voters as possible to outpopulate America's blacks, Jews, gays and Democrats.
Personally, though, what I find disturbing is not the number of kids they have but the patriachy which underlies their worldview... women can't go to school higher education or work, can't wear trousers or leave the house without a husband or brother and have their marriages arranged for them. It all comes from a theologically extremely dubious movement called ATI, headed up by a nutjob named Bill Gothard.
A woman calling herself Razing Ruth is writing a blog about having escaped from an attempt of her father to force her into a marriage within the movement. There's also No Longer Quivering, an online support group for women who have left the lifestyle. It can make for disturbing reading.
As Ma says, the biggest proponents of the Quiverfull/ATI movement are the Duggar family, who have their own reality TV show '19 kids and Counting' in the States. The Bates family have popped into a few episodes, and there's insinuations of marriages between their older kids and the Duggar's. Most episodes are available on YouTube...