Richman is the psychiatrist who explains the twisted inner workings of Norman Bates' mind at the end of the film. He's the voice of reason; he explains the seemingly random and perverse events of the film in clear narrative order. In fact, he's kind of a stand-in for Hitchcock himself; he's "directing" the sequence of events into an orderly narrative.
But is that voice of reason entirely reasonable? At times Richman seems to be a bit too enthusiastic about the unpleasantness he describes. He tells Lila:
RICHMAN: When Norman met your sister, he was touched by her… and aroused by her. He wanted her.
That seems like an uncomfortably explicit way to talk to a woman about her murdered sister, right?
Richman understands perversity; he explains it. But in doing so, he himself is stained with, and implicated in, that perversity.
You can look at the character of Richman in a few ways—he could be in this film to show that, no matter how scientific and detached you are, psychopathic acts affect all of us. Or, he could be there to show us that although Norman Bates' actions are gruesome, they're also completely captivating… whether you're a psychiatrist, a movie director, or a member of the audience.