For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
Act I is the beginning of the plot proper, when the characters commit to some course of action that's going to make up the story itself. When the narrator and Dupin come across the headline "Extraordinary Murders" in their evening newspaper, we see the first spark of interest that's going to drive Dupin to investigate the murders further. We also see the narrator's complete absence of a clue about who could have killed the L'Espanayes, a further sign that the case is difficult enough to be worth investigating for a marvel like Dupin. This first identification of the title mystery is when "The Murders in the Rue Morgue" really starts to lift off the runway.
Act II
The second act is traditionally the point when the plot is furthest from its solution. It comes at the moment when the newspaper reports that Le Bon has been arrested and Dupin decides to visit the crime scene. It's all an abstract intellectual problem until Le Bon, who Dupin apparently owes for some reason or another, gets arrested. Then, the whole murder thing gets personal, and Dupin starts examining the evidence directly. This up-close look at the crime scene leaves the narrator hopelessly lost – he doesn't see anything there that clarifies the problem at all – and Dupin is being unusually silent about the whole thing.
Act III
This act is the point where everything gets solved, so it makes sense that the third act begins at noon the day following their visit to the scene of the crime, when Dupin says he's solved the case. Dupin reveals to the Narrator that the killer's an Ourang-Outang (we were pretty surprised at this revelation, too). And it ends with the sailor's testimony, when he explains how the ape gets up to a fourth-floor apartment in the Rue Morgue. Dupin has told us what happened, his deduction has gotten independent proof, and all that's left is to sell the ape and laugh at the police.