How would the Rh factor cause problems with pregnancies?

Most of the times it does not cause problems.  It only becomes a problem when the mother is missing the Rh factor in her blood and the father of the child is Rh positive.  Even then it is only a problem after the first child is born.  The process is called Rh hemolytic disease of the newborn.  Essentially the mother will develop an immune response to any subsequent baby that is also Rh positive, if she is exposed to any of the Rh factors from her first Rh positive baby.  This used to be very common.  Now there is an antenatal (after birth) treatment of antibodies given to the mothers so she won't react to her subsequent babies.  To get a full description and explanation go to wikipedia.

If the baby is Rh positive and the mother is Rh negative, the mother will start developing antibodies to defend against the foreign substance (the Rh factor).  It's usually not a problem until the 2nd, 3rd, or later babies that the mother has developed enough antibodies to actually harm the fetus.

Then again, in the USA, the parents are tested for these things and the mother can be treated after each pregnancy to prevent complications.