This is an informal expression mostly used in British and Australian English. When you say that someone “talks nineteen to a dozen” what you mean is that the individual talks very fast in a non-stop fashion.
*The two old women sitting behind us were talking nineteen to a dozen. *I couldn’t concentrate because the couple opposite me were talking nineteen to a dozen.
No one is really sure about the origin of this expression. According to some people, the numbers refer to the words spoken. Before you manage to open your mouth, the person talking to you gets in 19 words.