My father bought me this book (thanks, Dad!) while I was in California recently. I wanted to share some thoughts with you from my experience as both a teacher and a student of Chess.
I'm about 100-150 pages into it, and I like this book. It's not for the faint of heart, however, and I haven't delved into much of the analysis and variations because I simply don't have the time. However, the book is quite readable, and Kasparov's effort of putting the games in context with a history and description of many of the many players and events surrounding the world champions is a welcome relief from the monotony of page after page of annotations and "informant" symbols found in comparable books of this level.
It should be noted however, that this book is not for junior students. In fact, I wouldn't recommend it to anyone under 1600 (perhaps even 1800), simply because there are other books out there that do a better job of catering to what junior players need to develop their game. That having been said, anyone who simply plays through the games and reads Kasparov's histories of the various world champions and their matches will be amply rewarded.
Some of the features of the book that I like:
(1) at the end of each chapter on each world champ, Kasparov summarizes with comments from other world champions regarding that individual.
(2) the moves to each game are printed in bold face so that it is much easier to distinguish the actual moves from the analysis.
(3) The analysis itself is insightful, and from what I have been able to ascertain, seems to be generally accurate. Of course, one should expect some errors as with any book. I'm not as much interested in variations and lines of analysis as I am in chess wisdom--general observations and maxims which I can put to immediate use--which is why I think that Bronstein's tournament book of Zurich 1953 is perhaps the greatest book on Chess ever written---certainly in the top 5.
(4) The language used in On My Great Predecessors is very well-thought-out and it's clear the writer took the time to express his thoughts precisely.
My chief reservation regarding this book is that it's difficult to tell what parts of the book Kasparov himself wrote (apart from the numerous "-G.K." quotes). I would like to think that Kasparov himself did a large part of the writing and analysis, or failing that, that he at least reviewed the analysis. It seems that the latter is true, although it's hard to confirm to what extent Kasparov himself was actually involved in the preparation of the text. It would have been nice if Kasparov's involvement had been clarified somewhere in the book. From the opening chapter where the author gives a one- or two-paragraph summary of each world champion, the author uses first person ("I see my style as...") when describing Garry Kasparov, suggesting that this paragraph (and perhaps that entire chapter) was written by Kasparov; however in the rest of the book the author attributes numerous quotes, including game analysis quotes, to Kasparov.
A comparatively minor issue is to what extent computers were involved in the analysis. A computer double-check is a good thing to have; however, anyone can load crafty or Chess Tiger on his PC and get good analysis from these 2600+ computer programs. In fact, the latest versions of Shredder are now over 2800! So when I buy chess books, I'm not looking for computer analysis but rather the insight---in English, not Informant symbols---which is unique to world-class players writing these books. However, it would have been good to see at least a blurb as to how computers were used in the analysis (e.g. what program, version, hardware, etc.).
In summary, from what I've seen so far, the book is destined to become a part of any Chess library, as important as the ECO's or ECE's. Once the whole three-volume set is out, it will probably become a standard reference work. I look forward to future volumes and editions.