Faust, Part I by Johann Wolfgang von Goethe. The scene in Auerbachs Keller — where Mephistopheles conjures wine from a table and Faust escapes on a barrel — is set in Leipzig, where Goethe studied as a young man. The whole play is transformed if you know the geography.
The Tin Drum by Günter Grass. Not set in Leipzig, but Grass received the Nobel Prize here and the novel captures the same postwar German reckoning with history that the city's museums address so directly.
Stasiland by Anna Funder. The definitive account of life under Stasi surveillance in East Germany, based on interviews with both victims and perpetrators. Read it before visiting the Runde Ecke; it provides the human detail that the museum's objects can only gesture at.
Leipzig — History, Art and Architecture by various. A solid general introduction if you want more background on the buildings, music history, and the events of 1989 before arriving.