"Breed" sounds like "breathe", so "breed their last" kind of sounds like "breathe their last". When one breathe's their last, it means they are taking their last breathe, essentially dying. Farmers are people that breed animals, or get animals to produce more offspring, typically for food. The joke is in the word "breed thier last" because it sounds sort of like "breathe their last" but breed ties into farming. 
OLD RANCHERS never die , they just breed their last .