Those phrases are called "refranes"("sayings"), and come from ancient times.
Here "el diente" means "la dentadura", the whole teeth, or better the tooth condition, like a general term. Like saying "the sea" naming all the Earth salt water instead of the particular Mediterranean Sea. Other better example: the Man named as the generic of the human race.
Old "sayings" jump country borders and languages, as you can see. Being "common sense" or "good manners", all "refranes"(sayings) have their "moraleja"(lesson). In this one we are advertised to not annoy the host with bad commentaries about the gift. Also to be graceful with our benefactor. Google returned me as a translation other different British "saying" that can complement the lesson: "Beggars can't be choosers"...
In Spanish there is another "saying" used to accuse somebody of not being thankful to his benefactor: "Muerde la mano que le alimenta". Here that person is being compared with a dog: "He bites the hand that feeds him".