There seems to be a great deal of confusion over exactly what inbreeding does as far as human genes go (keeping it very simple of course. The real deal is much more complex).I'll break it down for you all.
Every person has two strands of DNA; one a product of dad's two strands, and the other a product of mom's two strands. Being from different sources, these "copies" are not identical. Each of your two strands are complete- that is, they have all the information to code for a functioning human. However, neither of them are "perfect." If a segment of one strand is screwed up, the cell just uses the other strand. In this way, a number of people are carriers of severe genetic diseases, but they do not suffer from the disease. Although they have some segment of their DNA screwed up, the other strand covers it. DNA strands A and B can be absolutely riddled with errors and mutations, but as long as the screwed up sections don't overlap, you're just fine and dandy. (This isn't always the case, but for our purposes, its good enough).
When a person makes a baby with another person, generally its not an issue. While both Mom and Dad contribute DNA with flawed segments, generally, these flawed segments will not overlap, or at least there is a smaller probability that they will. In the case of incest, however, both Mom and Dad are from the same gene pool. There is a much greater chance of both having a flawed region that overlaps.
For example, let's say we have Genetic Disease BAD, which makes person sick for their entire life. BAD is caused by a flaw in DNA region M (region M must be flawed on both strands of DNA in order to make the person sick). Flawed M is found in 1 out of 100 people. Since Joe Schmuck must give either his healthy M or his flawed M to his child, the probability of his child being a carrier of flawed region M is 50%. However, the probability of Joe Schmuck marrying somebody else who also carries flawed M is 1/100. This means that, assuming Joe Schmuck picks his wife at random, the probability of his child having BAD is very slim (1/400 to be exact).
But in the case of incest, the chance is much higher. While the probability of having flawed M is 1/100 for the general populace, it is much higher in his family. Since the chance of flawed M being passed on from parent to child is 50%, a good number of the Schmuck family carries it (not 50%, due to some probability shenanigans, but certainly much higher than 1/100).
Let's say Joe Schmuck carries flawed M, and he marries his sister Mary Schmuck. Since Joe carries flawed M, at least one of his parents has flawed M, meaning that the chance that Mary carries it as well is 1/2. This means that the probability of their child having at least one flawed region R is 3/4, and the chance of the child having BAD is 1/8, a value which is much larger than the 1/400 for a random mating.
Short term, the chance of a bad disease being passed on increases dramatically. Long term, the probability of a Schmuck having flawed M will increase, meaning that the probability of a Schmuck having BAD will increase.
Will this destroy/weaken humanity with bad genes? No. Survival of the fittest. Healthy people produce more children than people with BAD. Will this impact and increase the chance of Schmuck children living miserable, sick lives? Yes. This was commonly found in royalty/nobility in the middle ages, where generations of inbreeding lead to a number of genetic diseases within the royalty. This also meant that extremely rare diseases (1/800 or 1/1,000) concentrated into these family lines, becoming much more likely to be expressed in the children.
If incest became legal, I'm of the opinion that children should not be allowed except through artificial insemination or adoption in order to reduce the chance of the child suffering from a genetic disease.