The exposition in this book is concise, intuitive and, for the most part, quite lucid. It's really tops for getting the larger view, relating key mathematical concepts to applications in physics. As an autodidact, I've found it particularly productive to use this book in conjunction with a more detailed treatment of some particular topic, e.g. tensors, representation theory for groups and the relationship between that and Lie groups and algebras. I've also found it enlightening to supplement this book with the typically more detailed and superb expositions on some topic in Frankel's The Geometry of Physics: An Introduction, Second Edition and in Wasserman's apparently lesser known but phenomenal Tensors and Manifolds: With Applications to Physics. With some foundation/supplementation, the book can profitably be used to solidify and extend one's intuitive understanding of these mathematical topics and come to understand how they are of use in physics. One can also use the book to identify weakness in one's understanding and to determine what else one needs to study to make further progress. In addition, Schutz provides solutions or hints to the exercises. It's a comparatively quick read and overall, quite enjoyable. Highly recommended for self-study but see the caveats below.
Despite my high praise, I think that this book is best used as a supplement to more thorough treatments of the math covered (mainly, differential manifolds, forms, Lie derivatives, Lie groups). Other books I have (repeatedly) used and highly recommend include Tu's Introduction to Smooth Manifolds (Graduate Texts in Mathematics), Lee's An Introduction to Manifolds (Universitext) on manifolds; Stillwell's Naive Lie Theory (Undergraduate Texts in Mathematics), Tapp's Matrix Groups for Undergraduates (Student Mathematical Library,) and Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction on Lie groups, etc. and Weintraub's Differential Forms: Integration on Manifolds and Stokes's Theorem, Darling's Differential Forms and Connections and at a more advanced level, Morita's lucid and concise Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) on differential forms.
Schutz states that the aim of "this book is to teach mathematics, not physics". In general, I do not think one's math should be learned soley from physics books (having experienced the inadequate job done on mathematics in typical general relativity and quantum mechanics books). This book is no exception despite its exceptional lucidity. The claim that one only needs reasonable familiarity with "vector calculus, calculus of many variables, matrix algebra ... and a little operator theory ..." is overly optimistic. In some narrow sense, it might be true that this is all that is required to follow the basic logic of the mathematical development, but to really understand the text, I believe some background in differential geometry, forms and Lie groups -- preferably acquired from math books written by mathematicians -- is required.
As I said, despite the caveats immediately above, I found the book both illuminating and enjoyable to read. In fact, I return to it quite often to refresh my memory on various topics.