
It shows that the dog has won at a certain level at Championship shows, as few dogs become champions it is a certain level of quality.
In my own breed a dog will get a stud book number if it wins it's Junior Warrant, gets a Challenge Certificate (the awards that count towards becoming a champion) or Reserve Challenge Certificate, or 1st or 2nd in the Open class (the class where Champions and dogs that have wonout of the lower classes compete) at Championship shows where Challenge Certificats are offered for the breed.
The stud book is where these awards are recorded, and I suppose having a stud book number shows the dog to be a good example of it's breed. It is not needed in order to use a dog at stud, but not many people would use a dog unless it had wins that would qualify it.