Economic Dynamics is the best way for an aspiring economist (early graduate student) to get tooled up to consume (and eventually produce) economic research. He begins the book with a fundamental intertemporal optimization problem that is easy to understand, but cannot be solved with most techniques that one learns as an undergraduate (Calculus, Lagrangian optimization...). This problem motivates the reader to learn the basic Analysis and Measure Theory in the following chapters.
He does a pretty good job at summarizing the basic results from Analysis that a graduate student should know in 19 pages. (It could have been a bit longer and included some material on inner-product spaces.) I found it better than chapter 3 of Stokey, Lucas and Prescott (SLP). However for someone who wants to do serious work in economic theory there is no substitute for learning Analysis from a textbook that focuses exclusively on it.
Using these basic results from Analysis, he introduces the reader to the fundamental ideas of Dynamics in Chapter 4 which is my favorite chapter in the book. Dynamics is an exciting subject that many undergraduate majors don't come across. He provides many famous applications of these methods from the literature including: Long and Plosser (1983), Hamilton (2005), and Quah (1993).
His introduction to Measure Theory and abstract integration is extremely intuitive (his illustrations on pages 161-162 help the reader visualize the concepts). His introduction to Measure-Theoretic Probability is very useful in many different areas of economics (from econometrics to finance to macroeconomics). He then applies all of these tools to stochastic dynamic programming in the following chapters.
At first I wasn't sure whether I liked his idea about using Python code, as many people in economics use Matlab or Gauss. Luckily he provides the equivalent programs written in Matlab on his website. Though I think he might be correct when he writes that Python is quickly gaining popularity. We'll have to see...
His website is a good resource for things related to his book, however I would also like to see the code he used to make the illustrations in the book, as well as a solutions manual to the practice problems in the text, as some of them are technical.
Bottom Line: if you have knowledge of multi-variable calculus, Linear Algebra, and undergraduate Probability and have some experience writing proofs (basic real analysis), this self-contained book is a great way for you to get tooled up in the methods necessary to understand Economic Dynamics.