Remember punnet squares from school? Looking at one of those will explain the problem with incest.
Essentially, many genetic abnormalities are attached to recessive genes. If both parents to carry that recessive gene, it will affect the child. If only one parent has the recessive gene, the child will most likely pass the gene to their offspring but they will not be affected by it themselves. Now, relatives tend to share most of your genetic abnormalities, so if you have children with a relative, you have an increased chance of having a child that have abnormalities.
So, for example, both of my parents happened to carry (but are not affected by) a recessive gene that causes a person's organs to be backwards. That means there was a fifty-fifty chance for their offspring to have backward organs. My sister is affected by this, whereas I just carry the gene. If we were to have kids with each other, they would be guaranteed carriers, and would have a whopping seventy five percent chance of having backwards organs. In turn, if our children were to have children, they would be pretty much guaranteed to have backwards organs.
Now, it isn't necessarily always that simple, but that basic effect makes inbreeding dangerous. Examples include the
hemophilia in European royalty. An even more extreme example is Spain's
Charles II, who's family had been inbreeding since the medieval ages until they were rife with crazy problems. He was barely able to function, let alone breed, and his incompetent reign only ended when he died without an heir, causing
The War of Spanish Succession.
That all being said, the short answer for why there isn't much support for incestuous marriage is because... there isn't much support for incestuous marriage. Homosexuality is relatively common, but incestuous relationships are not so common, and incestuous relationships where the couple want to marry is way rarer.
