Because it assumes that the shelter dog and the dog purchased from a breeder are the same, with equivalent traits. This is only very rarely the case. Most often, they are apples and oranges.
Now if someone is just looking for a sound family pet and doesn't have particular requirements for breed, size, color, etc... then certainly they may be able to find this dog in a shelter. If they are looking for a sound family pet that is also a purebred of a specific breed, their chances drop but it is still possible.
If, as has been said previously in the thread, someone is looking for a dog for a specific purpose... competition in any number of sports, police work, SAR, guide work, service work, show, then the chances of them finding such a dog in a shelter are very slim. If they want documented pedigree and known ancestry on the dog... which isn't just for ego but can provide valuable insight into any health or temperament problems that may be present in the bloodlines and thus arise in the future... then the chances of finding that in a shelter are pretty much zero.
So bottom line, for many pet owners the shelter is a viable option. For pet owners looking for something specific, it often is not. For people looking for dogs for sport or work, they'd have a better chance finding a needle in a haystack in most cases. Thus, there is really no competition. The shelter doesn't offer what some people want, and breeders are the only option. Likewise, the breeders with proven track records of producing quality dogs that can succeed in a variety of endeavors are typically not the ones whose dogs end up in the shelter.
So it is absolutely NOT correct to say that when they buy from a breeder a shelter dog dies, because what they want isn't going to be found in the shelter.
When my children were young all of our family dogs (pets) were rescues or strays that wondered into our yard. They were wonderful and loving and fulfilled the purpose for which they were needed.
Move ahead years later and when I began looking for a dog for me and my purposes I went to a breeder.
I have been involved in rescue and a bit of fostering for years. I pulled, dropped off & picked up dogs from the vet, transported, fund raised, joined groups, took and posted pictures on websites, and did home checks for rescues out of area that were adopting to people in my area. For a couple of years I sat on the Board of Directors for our local humane society. Besides the local HS I also did work with our county animal shelter and was part of a team that helped bring it from a high kill to low kill shelter. I know the highs of saving and the heart break of dogs being euthanized.
I do not go along with that saying about buying a dog from a breeder means a shelter dog dies. I will not let anyone lay blame on me for any dog dying. I could very well turn around and claim that because of them adopting only 1 dog from a rescue or buying only 1 dog from a shelter instead of two that they allowed a dog to die.
I have been a CGC Evaluator for a long while and I would say about half the dogs I have passed were full blooded and about half were mixes. The same ratio probably pretty close to those I have failed. So for pets for families this testing goes along with my previous statement that a good family pet (or even a pet for an individual) a rescue is a wonderful choice. I always recommend a rescue over a shelter dog for a family with children as most rescues have had the dog longer and a good many have had them in a foster setup of some type.
I am also very involved with Assistance aka Service Dogs for people with disabilities. I know how hard it is to find the proper dog for the work. I have evaluated dogs in shelters and found many that would make a great pet but never one that I thought was suitable to be pulled as a SD Candidate. I know many people who claim to have done so and I have seen some of the dogs during and after training.
I am in the process of looking for a new pup for myself. It hasn't been easy as I have a want list that some items I could work around while some I hold as a very high priority. For this I need to go to a breeder who I can trust to be looking out for my needs as much as finding a good home for their pups. I need a breeder who knows about the parents and the grandparents and how siblings and others in the family lines have turned out. I need more than finding a pup that will grow up to be friendly with my guests and loving to family. I need a pup that I can raise and train to be my working partner in several venues.
As Chris mentioned track records for working dogs -- I need a pup from lines with certain proven track records. I need a dog that will grow to at a least a minimum height. I need a dog that has a very good chance to have a good work ethic and solid balanced temperament.