In the real world amanuensis has come to mean a special secretary. In Anathem, the term applies to an Avout who makes a written recording of an events or dialog. The avout's only media are chalk, ink and special new matter recording disks.
Fraa Orolo appoints Erasmas as an amanuensis at the start of the story for his interview with artisans Flec and Quin just prior to Apert. Erasmas finds the interviews ridiculous but Orolo wants them recorded.
Much later, Fraa Jad tells Erasmas he is to be an amanuensis for Jad's interview with the Gan of the Daban Urnud, which takes place in one of the narratives of their attack on that ship. Jad details that this role is important, and different from the ship's own electronic recording devices, stating:
"An amanuensis is more than a recording device. An amanuensis is a consciousness-bearing system, and so what it observes in its cosmos has effects in others, in the manner we spoke of at Avrachon's Dowment."
This refers to Polycosm theory, where, in the book, observers can affect the universe and other observers, and cause information transfer to other parts of the Polycosm. This is an unusual mixture of the two main real-world theories of quantum mechanics, namely the Copenhagen interpretation where the observer's actions cause a waveform to collapse to a single state and the Many worlds interpretation which is much closer to the Polycosm theory of Anathem.
In a literary sense, Erasmas is also our amanuensis for the entire story. He observes it, and we readers perceive only the worldline that he observes.