Stepping back farther into history, suppose someone came back from a distant land with an exotic chicken breed. Only a few made the voyage. Then you only have a few to start with. Entire breeds have started this way.
In other cases, there are only a handful of individuals to work with, at a breeders disposal, that would be considered pure. Only a few of those have the desired traits. The best way to fix traits, as illustrated above, is to "line breed" (the term we use when inbreeding works out well) . Inbreeding fixes traits, that means it sets them, as in affixed, it doesn't fix them as in improving them. If bad traits are there, as they were in some of the royal families that gave rise to the notion that inbreeding was a bad thing, the bad traits will be magnified just as the good traits would be.
There are many families of chickens that are highly inbred, and nothing is broken, so nothing needs to be added. Doesn't take much to revitalize a line if it gets too close, and often takes further inbreeding to weed out any bad traits that were introduced with the fresh genetic material.
I personally have chickens that are highly inbred, and there are hens within my line that are hatching chicks from their own eggs as teenagers. They have been hurt in no way, because I and others have been careful not to breed anything showing the slightest hint of bad traits. The bad traits simply aren't there, and with no new genetics, there is nowhere for them to come from besides a mutation, which would be weeded out if it occurred.
People have the notion that you can't breed a rooster to his kin, when in reality, there are commercial production chickens that have been carefully linebred for close to a hundred years with no outside introductions.