What does this mean Have a dog in this hunt?

This is an idiom that expresses the level of concern for a situation. The idiom is commonly heard in the negative as in, when confronted by a situation, you express anopinion but then follow it up with "but I ain't go no dog in this hunt", meaning that although you have an opinion you don't have anything to lose or gain so you can accept whichever outcome.
It comes from the hunting culture, probably in the mid or deep south where everyone would show up for a bird hunt with their dogs. Dogs were retrievers for their masters and would bring back the shot game to their masters. Of course if you didn't have a dog in the hunt you wouldn't get any game. It could also refer to a larger game hunt such as a fox hunt or bear hunt where the stakes were a little higher. In this situation your dog could be harmed by the game and your level of concern would go up because a good hunting dog was really valuable. Remember the old joke: Man tells his friends: "My wife got mad at me for staying out late hunting again. She told me it was either my wife or Sam, my huntin' dog. I surely am gonna miss that old gal."
This is actually an amalgamation of two separate expressions, "that dog won't hunt" and "i have no dog in this fight".
"That dog won't hunt" means what you're attempting to do will be met with failure, as in "no matter how many times you run the race, that dog won't hunt". The origin of that phrase does originate in southern hunting culture. It relates to coon dogs that don't have a good sense of smell, thus not being a good hunting dog.
"No dog in the fight" means you have no concern for the outcome of a given situation. An example would be, "I'm a Braves fan. Red Sox-Yankees... I got no dog in that fight". This phrase has origins in the dog fighting culture.