4 ideas
Dupin, working from newspaper reports and one visit to the scene, finds that a window nail is broken so the sash only appears fastened, notes the superhuman strength and agility required and the non-human hair in the victim's grip, and concludes the killer was an escaped orangutan. He places a newspaper advertisement for a lost orangutan that draws in its owner, a sailor, who confirms the account.
Dupin argues the police are baffled because they mistake the unusual for the hard to understand. The strangest features of a crime, such as the voice no witness could place or the pointless brutality, are the most useful clues because they rule out ordinary explanations. The right question is not 'what happened?' but 'what happened that has never happened before?'
Each witness names a different foreign language for the shrill voice, and each names one they do not understand. Dupin reads the pattern of disagreement rather than any single account. When independent observers disagree in the same way, that shared feature, here 'no recognizable language', is the real evidence, and the details each observer adds are noise.
Poe opens by arguing that the true analyst is not a calculator but someone who observes and infers what others overlook. At whist, the skilled player reads opponents' faces, hesitations, and discards, and so models their minds. Dupin shows this when he follows the narrator's silent chain of associations and states the thought the narrator has just reached.