Dr. Fletcher is Dr. Railly's boss at the county mental hospital. His role in life—or, at least, his role in the movie—is to be a foil for Dr. Railly. That is, he is a character who provides contrast for Dr. Railly and the changes she undergoes.
Throughout the story, Dr. Railly slowly begins to lose her sense of what's real and stops judging her decisions on the basis of rationality. In her own terms, she is "losing [her] faith" in psychiatry and science. In contrast, Dr. Fletcher is an unwavering shoulder angel for Dr. Railly; only instead of being a voice for good or evil, he is the voice of reason.
We first see this when Cole is in the mental hospital. Railly doesn't restrain him as she should, and Cole makes a break for it, injuring several people in the escape attempt. Dr. Fletcher rightly and reasonably talks with her and points out she made a mistake.
When Cole does escape, Fletcher can't believe the orderly's story because a fully restrained patient shouldn't have been able to climb through a one-foot-by-one-foot air vent. The idea that the orderly may have been telling the truth doesn't even cross his mind.
Later, when Dr. Railly is on the brink of believing Cole's story, she calls Dr. Fletcher to talk to him. Despite her claims that Cole knew about the boy hiding in the barn, Fletcher soundly asserts:
DR. FLETCHER: No way he could possibly know that. Kathryn. You're a rational person. You're a trained psychiatrist. You know the difference between what's real and what's not.
Unfortunately, he fails at his job of being a rational-trumping shoulder angel. Railly doesn't take him up on this advice, and what's real and what's not becomes less clear for her than it does for his ever-rational mind.