As Dupin exhaustively explains, he thoroughly examines the room where the murders took place. Although the police have already done so, Dupin goes over the entire apartment again and discovers no secret or hidden entrances. Both doors out of the room are locked. The fireplace chimneys are so narrow that not even a cat could get through them. The two windows are locked from within, and there is no way the murderer could have climbed out of them and locked them from without. Dupin also excludes the idea of a supernatural occurrence.
However, from a very careful examination, Dupin sees that the assassin could have escaped through the window overlooking the bed. The window is designed to spring back into place, and what looks like a whole nail holding the window in place is only a portion of a nail, so it doesn't really lock the window. As Dupin explains,
Pressing the spring, I gently raised the sash for a few inches; the head went up with it, remaining firm in its bed. I closed the window, and the semblance of the whole nail was again perfect.
The riddle, so far, was now unriddled. The assassin had escaped through the window which looked upon the bed.
This is a very arcane explanation which is difficult to follow, but the point is that a minute and careful examination of a crime scene is necessary: the police did not look closely enough to discover what Dupin discovered.