As Rothfuss explains in both the Forward and Endnote, this isn't a typical story, and it only really works if you are already familiar with Kvothe's story, not because Kvothe himself appears in the book, but because the understanding of the world and Auri's introduction and role in The Name of the Wind is necessary for The Slow Regard of Silent Things.
In this book, we watch Auri over the course of several days. We see how she spends her time in the Underthing, a maze of passageways and long-forgotten rooms below the University. Each day plays out differently as she wakes up inspired to perform different actions. These can be anything from days where she must hunt around the Underthing exploring and looking for new items, days where she must fix problems, days when she must gather food, or days when she needs to make new things, or, unfortunately, days of utter sorrow where she can't even get out of bed.
Throughout the entire seven-day period, which takes place between chapters seven and eleven of The Wise Man's Fear, Auri finds herself not only attempting to accomplish the various tasks the day apparently demands of her, but also trying to fix all of the little things that are off in her small world. As she moves from room to room, she examines everything and when she feels like one item in the room isn't happy with its placement, she does everything in her power to move it or the trinkets around it in order to bring peace to the item and room again.
Given how little we know of Auri's past, how far her training went, and what event caused her to go into seclusion, there were many times during this book where I couldn't decide if she had some insight into the nature of the world, or if she was suffering from a form of OCD or, most likely, some mix of the two. As a result, reading as she goes through her daily process is both a wondrous and sad journey, especially during the times when things just don't want to go right for her.
Both The Name of the Wind and The Wise Man's Fear showed that Rothfuss has a great command of the language, and a way of weaving his tale to keep you enthralled. What The Slow Regard of Silent Things shows is that his skill is even greater than that. Rothfuss shows Auri's slightly-off view by not just describing what she does, but by also getting the readers into her head and having her unique way of speaking and thinking permeate the entire story.
While The Slow Regard of Silent Things itself only has a slightly fantastic theme to it, and should be enjoyable even to readers not interested in the genre, the requirement of having gone through the first two Kingkiller Chronicle books in order to set the stage means that, most likely, only fantasy fans will get the full impact of this book.