Good manners will open doors that the best education cannot.

It means that if you have good manners, you will have better opportunities than when you have a really good education. For example, Person A is really smart and Person B is not that smart. Person A is rude and Person B has good manners. The phrase means that Person B has more opportunities than Person A, such as, at jobs. I do not think that the phrase is true because, who gets a better job when you are not educated than when you are educated?