Monsieur Defarge is hunting for any written evidence that Dr. Manette might have produced before he lost his mind and forgot completely what had occurred which resulted in his lengthy internment. He is eager to find this because his wife, as the sister of the family who was so wronged by the Evremondes, has pledged herself to gain vengeance on the Everemondes. Such a letter would give her ammunition against Charles Darnay, and allow him to be convicted and executed as an enemy of the people.
Note how this event is foreshadowed in Chapter 6 of Book the Second, quite ironically, by Charles Darnay himself. He tells the story to Dr. Manette and Lucie about the Tower of London - the "double" of the Bastille in terms of its function as a prison. H remembers how workmen came across a secret dungeon which had remained forgotten for many years. One prisoner had written "Dig" on the wall, and underneath a tile were found the ashes of a paper. The prisoner had written something down and concealed it from his imprisoners. Note the reaction this has on Dr. Manette:
He had suddenly started up, with his hand to his head. His manner and his look quite terrified them all.
This clearly suggests that on some subliminal level Dr. Manette had likewise at one time produced a letter which he had hidden in his cell, which Defarge finds and is so tragically used to condemn Darnay, Dr. Manette's son in law.