I think many places have laws regarding incest and who can marry who, and this is the reason. Where I live, it is unlawful for two people who are more closely related than second cousins to marry, but they will make an exception for first cousins if the woman is past child-bearing age. (Yuck.)
Some people call it "inbreeding" and yes, there is a greater incidence of birth defects, and often the child has a strange look about them. Often the child has mental handicaps or lower intelligence. There are also cases where the child is totally normal. With all the risks, it's better to control yourself and avoid these problems.