The text that you quote is quite difficult to understand on its own. It helps to look at a summary the plot of "Bartleby, the Scrivener" by Herman Melville:
Bartleby answers with what soon becomes his perpetual response to every request: "I would prefer not to". To the dismay of the lawyer and the irritation of the other employees, Bartleby performs fewer and fewer tasks and eventually none, instead spending long periods of time staring out one of the office's windows at a brick wall.
To rephrase the section that you have quoted:
When he is asked to do pretty much anything, he replies "I would prefer not to".