Holden is not a stable personality. We don't know much about his psychological state before his brother died, but afterwards he was a little emotionally unbalanced. He did not process the death of his brother very well (if anyone ever processes something like well) and the night Allie dies he sleeps in the garage and proceeds to break all the windows with his bare hands. In addition, Holden has been getting drubbed out of all the nice, expensive, exclusive schools his parents keep getting him enrolled in. For these reasons, his parents want to have him psychoanalyzed to see if they can determine what is "wrong" with him. They don't seem to think that sending him away on his own for long periods of time has anything to do with his inability to get along in life. Their solution is to turn to mental health professionals instead of perhaps giving him the attention he needs.
In The Catcher in the Rye Holden's parents want to have him psychoanalyzed because he had an emotional breakdown after his brothers death. When his younger brother Allie died, Holden lost it and broke all the windows in the garage with his bare hands, to the point where he could no longer squeeze his hand. We can still see that he is effected by Allie's death. He often mentions how he brother was a nice kid, and how he had a great future ahead of him. He also still carries his brother's glove with him.