The purpose of the flu shot is not to get your sick.
It's to train your immune system to develop antibodies against the flu strain without making you sick, thereby preventing you from catching that particular strain. (Or those strains if the vaccine is based on multiple viruses).
Flu shots use dead or severely weakened (in nasal spray vaccines) flu viruses. That way, if the weakened virus does happen to make someone sick, it's so minor that they don't really feel anything.
People who get sick after a flu shot usually either get a different strain of flu virus (really, how many of us can tell the difference between flus), caught the strain in the shot before the shot could take effect, or the shot did nothing for them (could have been too weak or diluted, or their immune system just didn't respond to it) and don't build the immunity.
It's relatively rare that a shot based on a weakened, rather than dead, virus causes the flu. What people usually experience are flu-like symptoms which are actually a negative reaction to the serum in the shot. But usually that happens with people who already have weakened or compromised immune systems for whatever reason (like they're already fighting something off but don't realize it).
Seven years straight with flu shots, and I think I had one flu, months after the shot. And I've had flu shots before that and don't remember ever getting sick from one.