How does Sydney manage to rescue Darnay? Explain this thoroughly. (From Book 3 Chapters 11-15)

Let into the prison by John Barsad, Carton visits Darnay an hour before his execution. He convinces Darnay to swap clothes with him and drugs him to put him to sleep. John Barsad enters the cell to drag Darnay to safety, and Carton remains in the cell to die in his place. A gaoler takes him to a waiting room, where various other prisoners mistake him for Darnay and greet him. A young woman accused of plotting recognizes that it is not Darnay but keeps his secret and asks to hold his "brave hand" on the way to the guillotine.

A coach holding Doctor Manette, Lucie, little Lucie, Mr. Lorry and an unconscious Charles Darnay disguised as Sydney Carton (and holding his papers) pass safely out of Paris.