The idea is not as counter-intuitive as it sounds. Just as every other organisim out there, the purpose of a virus is to replicate and ensure that the strain lives on. With the widespread availability of the flu vaccine, some strains of the flu are unable to go from host to host and replicate themselves. Those strains that cannot replicate themselves are gradually eliminated from the virus population.
Viruses mutate just like any other organism and, in the case of the flu, the mutant strains are not stopped in their tracks by the flu vaccine. Thus, we are left with resistant strains of the flu floating around. These resistant strains tend to be stronger and more virulent that the virus knocked out by the vaccine.
So, the introduction of the flu vaccine, meant to shield as many people from the flu, has, instead, contributed to the development of stronger strains of the flu that are resistant to the current vaccine.
Economists refer to this phenomenon as the law of unintended consequences and it is (virtually) a universal truth: any attempt by a simple system to regulate a complex system generally leads to unexpected (generally negative) outcomes. The war on drugs brought us more violent street crime, overcrowded jails and a generation of black males whose contact with the criminal justice system is scandalous; and MADD's rampage has made criminals out of ordinary folks who were in the wrong place at the wrong time.