Getting rid of all the mucus in your body is a flatly ridiculous claim. Mucus is an important part of your barrier against invading microbes in the nose, mouth, throat, and respiratory tract, and is constantly produced by the cells that line these spaces. I therefore would not trust any other information from the same source.
Which is not to say getting rid of dairy will not improve your health. I've gone a week with no dairy and not noticed anything one way or another, but experiences differ.