Why do they say that pure breeds are weak?

In general, purebred dogs have a smaller gene pool than a mixed breed would come from.  This gene pool increases the chances that a recessive gene would become a dominant gene, thus increasing the likelihood of genetic faults (hip dysplasia and others of that ilk.) 
 
Since a mixed breed has more genetic diversity to call upon, they can be considered to be stronger (in general) than a purebred.
 
Purebreds are more likely to have been bred for specific purposes.  Hundreds of years ago those who bred knew nothing about OFA or CERF certs, and sometimes the dogs that were the best hunters, retrievers, and herders carried those genetic faults with them.  Since those dogs were preferred due to their abilities, those faults were made more pronounced and spread across the gene pool. 
 
Now that we know some things about those genetic disorders, we can test for the presence in the prospective brood bitches and sires, but we cannot test the genes themselves.  Since we cannot test the genes, we are ruling out that the gene is dominant, but we cannot rule out that the undesired trait/defect is not recessive.

I understand what Xerxes is saying but as for the mixed is better then a purebred.....ha ha hahahahahaha........ha ha!

It depends on what kind of purebred dog we're talking about.

A purebred dog bred by idiots who did no genetic testing or health certifications and know nothing about the pedigree of their dogs is very very likely to be very very unhealthy. However, a purebred dog bred by a reputible responsible breeder who has paid very close attention to the health of both sire and dam and their pedigrees has a good chance of being pretty healthy.

A mixed breed dog that was created au naturel has just as much chance of being unhealthy or healthy depending on the dogs that did the deed. Two dogs with genetic problems are going to have a high chance of throwing pups with more genetic problems. It doesn't matter if the parents are pure or mixed. Genetics is just like that.

I prefer to break it down into "random bred" and "purpose bred" for this type of discussion. Dogs which are matched with care taken to increase the likeliness of soundness, will almost certainly have a better shot at being sound than a pup that is the result of a random breeding, whether purebred or mixed breed.

Dogs which are "outcrossed" have slightly better odds of not inheriting a double shot of a recessive gene that is unknown to exist. Inbreeding lines known to be clear, however, can set soundness in the line, while matching known carrier lines with non-carrier can continue excellent traits that the carrier lines offer, while increasing the odds that a non-carrier will be produced with those same desireable traits. My own pup Ted is the product of such a breeding - single gene carrier to non-carrier - I do not yet know his status but he will remain un-neutered for that reason until I know. This is possible with a few of the eye disorders that affect Border Collies, plus the MDR-1delta mutation, as there are now DNA tests to detect the genetic status of a potential breeding animal - whether the animal carries two genes, only one, or is not a carrier at all.

So it has nothing to do with whether the dog is the same breed or not, but whether the two parents carry the same genes for soundness issues.

ETA: It is primarily the designer mutt breeders these days who have spread the hype about purebred dogs being intrinsically less sound than a "cross breed" - it should be pretty obvious what their motivation is in getting people to beleive that purebreds are somehow inferior to their overpriced puppies.

A mutt has less chance of matching up two defective genes than any purebred (unless the two parents are from closely related breeds). The genes that cause problems in one parent's breed aren't likely to be the same ones causing problems in the other parent's breed... so each parent's good genetics covers up any defective, recessive genes from the other.

Carefully bred purebred animals have advantages, though. The breeds have certain qualities and certain abilities for which they were created; their adult size is known; their temperament can be loosely predicted. A mutt can't be predicted in that way; and he probably won't be a good "specialist" working dog like so many purebred dogs are. The genetic problems can be almost eliminated, with care, though there'll always be a chance of some unknown mutation coming to light.

It's when people are stupid about just churning out puppies with a certain label that things get to be really sad and really stupid--purebred dogs with their lifespans cut in half, dogs with defects that could've been prevented if people cared about more than money.

People who deliberately breed mixed-breed puppies can really only say that the pups' health will be better in the first generation, with each parent a purebred from the two breeds--and that first generation is the least desirable to the breeder, since the pups' appearance is all over the board between the two breeds and can't be predicted. In the second generation and beyond, the pups have genes from both breeds, and thus defective genes from both breeds; and they are just as prone to "matching up" two recessive genes as any purebred animal. What adds to this problem is that a lot of people who do this sort of thing are just doing it for the money.

It is also important to note that while Mixed breed dogs tend to be healthier initially, the mixed breed bag also means that you might not look for or detect genetic health problems early enough to make treatment effective.  If you have a specific breed, you, and your veterinarian, are looking for evidence of common problems and ailments associated with that breed.  A good breeder has been screening for those problems right along in their breeding program.  With a mixed breed dog, you can get the unexpected manifestation of ailment or defect.  And in many cases, the actual breeds of a mix are only known for sure on one side, so there is an entire genetic map that you are unaware of.  In mixed breed pups, you can account for the brood bitch, but the sire is a complete wild card.  As it is also possible for a dog to mate with more than one male animal while in heat, the litter may reflect more than one sire-and consequently, unless you are willing to go for expensive genetic testing, you don't know for sure what to expect once again.
 
In general, it seems that the strongest genetic traits come out in mixed pups, and you find the recessive traits much less often that you would in a purebred dog, because genetically, many traits are linked to other traits in the DNA chain. 

My last dog was a mixed breed (Lab/Boxer/Pit) and she had a host of health problems her entire life.  We managed her skin and ear issues for years. My current dog is a purebred and comes from a very, very reputable breeder. She will be seven years old in a few weeks and has never has any health problems.

ANY animal can develop health issues not related to age or injuries.  Just like with people, it happens.  Sometimes it's luck of the draw, sometimes it is an obvious gentic flaw. FWIW, some genetic flaws have been turned into things people want, i.e. color (all white), different head shape, long backs, etc.

I also wanted to add that I'm a fan of purebred dogs precisely because of the predictability of temperment, size, and other associated things with the purebred.  I also like the fact that when you adopt from a breeder, you get a great resource in that breeder, and someone that knows what to expect from his/her pups.  (ie: is it normal that he....???) 

I also wonder if when the AKC started putting more regulations on breeders that were producing and registering many more puppies than the average reputable breeder, such as requiring them to have a DNA sample of frequently used stud dogs on hand (this made it harder to hang pedigrees on dogs that were not actually the descendents of that line), the puppy millers began to cross between the breeds they have on hand and publisize the alledged "hybrid vigor" that resulted from the outcrosses.  This way they could sell the dogs for high prices without having to register them.  Although hybrid vigor exists, a mixed breed dog is hardly hybrid.  If a dog crossed with a wolf does not qualify as hybrid, neither does a cross between two dogs.  I think it was based in fact, but blown way out of proportion by people who just wanted a cut of the profit.