Well some shots do work like that. But you usually don't get symptoms unless you are immunocompromised and it's a live (but weakened) version of the virus.

They were just a little wrong, “Flu shots give you a flu”.

There are 2 types of these shots essentially:

1. the pathogen is put into some other thing that creates stuff that fights against said pathogen. That stuff is then extracted and given in the shot.
2. the pathogen itself is processed and given to you. This causes your body to make stuff that fights against the pathogens. Your body then vaguely remembers how the pathogen felt and hence, increases the reaction your body does to any attack from a similar pathogen that comes the next time. This is the one corresponding to the above quote.

Of course, if your immune system is weak, the processed pathogen can be enough to give you quite a bit of a problem.