Well it isn't just this culture. I worked with a couple who were married and had two kids. I found out after a while they were cousins. They were open and honest about it, never tried to hide it at any point.
They did go for lots of tests during both pregnancies, they understood the risks (not sure what they would have done if something was found though)
They were both White, English, and decent upstanding people. Legally they haven't done anything wrong. In the animal world I'm sure many animals cross breed with their own siblings etc.
Personally I would never form a relationship with a cousin etc, but they did and they were fully aware of the risks and dealt wiht them accordingly.
They were educated, this is the only difference between the people in the OP and the ones in my post here, lack of education and/or heads buried in the sand.