It is to emphasize the word "
fuck." It comes from the shorter version of "give a
fuck" and then "give a flying
fuck." It means "I don't care" but with emphasis. So it is an emphatic "I don't care!" Or also, emphatic "I could care less."
"I
don't give a flying dog
fuck if you want to go to the beach, we are NOT going!"
"I don't give a flying dog
fuck if you like driving, I am doing the driving!"