Three honest reasons, none of which require a conspiracy.
First, immune responses feel lousy sometimes. Your body is building protection. That can look like being "sick" even when no pathogen is winning.
Second, flu season overlaps with every other crud that circulates. Rhinoviruses, COVID-19, RSV, and friends all produce fever, cough, and misery. The CDC explicitly warns that someone can get sick with another respiratory virus and mistake it for flu, or blame the vaccine for a bug they were already incubating.
Third, flu vaccines are not force fields. They target specific influenza viruses. If you meet a strain that does not match well, or you get exposed before immunity ramps up, you can still catch influenza. That is exposure plus timing, not the shot infecting you. Vaccination can still make a breakthrough case milder, which is a separate benefit from "I got sick anyway."