Every time you reproduce there is a small probability of a mutations. Most of mutations are dangerous. But often since the baby will have one strand of DNA from one parents and one from the other, the probability of the two strands have the same mutation is really low. So the children will appear with no defects (I am grossly simplifying, some times a single strand mutated is enough to have a deadly mutation).
Now suppose a person A has a mutation in one strand of the DNA.
He has 4 kids, two males: M1, M2 and two females: F1, and F2.
And now suppose that two of them M1 and F1, carry the mutation. The mutation appears as a different base in one of the chromosome. Now they reproduce, and make 4 children: b1, b2, b3, b4. Statistically each of them will have the mutation on both strands with a probability of 1/4. On only one of the strands with a probability of 1/2, and no mutation at all with a probability of 1/4.
So if a person has a silent mutation, and his kids interbreed they will make the mutation appear in full force with a probability of 1/4.
This says nothing respect to what the effect of the mutation is. It could be positive, or it could be negative. From a human perspective it is not acceptable to have a probability of 1/4 for a children to have a mutation (not to mention the 1/2 to carry it in a silent way). But from Nature point of view this only rises the probability of having mutations. Something which Nature can still cope quite well (still because there is a mutation threshold after which the number of viable offspring is too low for the species to reproduce). After all the mutations that lead to birth defects has often limited effect (species-wise), while a mutation that causes a neutral or a positive mutation can have a positive impact on the species. So you would not interbreed human beings. But in the past it has been considered acceptable to interbreed some animals looking to extend specific traits.