Painting by Numbers is a system where a picture is divided into shapes, each marked with a number that corresponds to a particular color. You paint in each shape and ultimately the picture emerges as a finished painting.
The paint by numbers approach is often ridiculed as being simplistic, uncreative, and formulaic. I believe it's helpful in getting across the concept that a painting is built up through multiple shapes of color. These shapes often don't make sense individually, nor look like anything "real", but put together as a group they create the image.
The next step in developing as a painter is to learn to see such color shapes for yourself, without the aid of a printed diagram. Completing a paint by numbers project helps you learn to analyze a subject and observe areas of color. It helps you move away from focusing on what the finished subject will look like to looking as small areas and what color these should be painted.
"'Painting by numbers' may not be as egregious a pursuit as one might imagine. Leonardo himself invented a form of it, assigning assistants to paint areas on a work that he had already sketched out and numbered."
-- Bülent Atalay in his book Math and the Mona Lisa: The Art and Science of Leonardo da Vinci