Why does Poe have the Prefect leave before Dupin begins his explanation to the narrator concerning the solving of the crime?
Poe's main reason for having the Prefect leave before Dupin explains how he recovered the missing letter must have been to emphasize that the problem and the solution are Dupin's and that he is the hero. Dupin may have had incidental reasons for waiting for the Prefect to leave--although he didn't have to wait very long anyway, because the Prefect looked "absolutely thunderstricken" and left without saying a word.
At the beginning of the story, Monsieur G- is seeking help but at the same time is laughing at Dupin and ridiculing his suggestions, including
"Perhaps it is the very simplicity of the thing which puts you at fault."
Prefect G- says
"What nonsense you DO talk!"
"Oh, good heavens! who ever heard of such an idea?"
Dupin may have been a little piqued by the Prefect's ridicule and was taking pleasure in keeping him mystified. Since the police official had ridiculed his suggestions before the letter was recovered, he might have been inclined at least to reject them afterward and attribute Dupin's success to pure luck.
It might also be to Dupin's practical advantage to have Monsieur G- think of him as some sort of a wizard who could produce such amazing results by magic. No doubt the Prefect would bring Dupin other problems in the future, some of which might also entail lucrative rewards. Dupin, like Sherlock Holmes, is an amateur detective, and as such he needs good relations with a police official in order to have problems brought to him as well as in order to have an aegis under which to conduct his investigations.
Arthur Conan Doyle repeatedly admitted his indebtedness to Edgar Allan Poe's tales of ratiocination. Doyle's Sherlock Holmes often is seen to follow Dupin's example in letting the police take credit for solutions to mysteries and also in frequently refusing to explain his "methods," including his practice of making elaborate deductions from simple clues, in order to amaze and mystify people.
“The Purloined Letter” is Dupin’s story, not the Prefect’s. Poe uses the Prefect only to introduce the problem, explain the elaborate attempts to solve it and the large reward offered for its recovery, and then leave it with Dupin. When he returns a month later he repeats what he said at the first meeting:
“I would really give fifty thousand francs to anyone who would aid me in the matter.”
“In that case,” replied Dupin, opening a drawer and producing a checkbook, “you may as well fill me up a check for the amount mentioned. When you have signed it I will hand you the letter.”
This occurs on about the seventh page of a fifteen-page story. The Prefect appears “thunderstricken” and leaves without saying a word. The remaining eight pages—slightly more than half the story—are devoted to Dupin’s narration of his thoughts and activities as he went about recovering the stolen letter. Poe gets rid of the Prefect quickly and finally because he wants to emphasize that the problem, or conflict, as well as the resolution are Dupin’s.
The essential problem is not to recover the letter but to find out where it is. Evidently this is something the Prefect never learns, but the story would be totally unsatisfactory without that full disclosure.