I am doing my graduate study in Applied Statistics. This semester my professor told me to take Real Analysis and the professor who is teaching Real Analysis is using Folland's Book entitled "Real Analysis: Modern Techniques and Their Applications." The beginning chapters are very abstract for me. The title of the book is ... Modern Techniques and Their Applications, so is this book really applicable to Applied Statistics later on or the application of this book is just for theoretical analysis?
Real analysis is absolutely not an applied subject, by which I mean it's improbable that you'll solve any practical problems, or even impractical models of practical problems, during this course. It's the language in which probability is developed, though, so as a statistician you'll probably be expected to know some probability theory, which is why analysis was recommended to you.
Applied mathematics is just a collection of subjects that are traditionally or usually associated to applications. It does not mean that everything in applied mathematics must be "applied to something". For example, differential equations and graph theory are usually considered as part of applied mathematics, but they can be as theoretical and far from applications as anything can get. Real analysis is not among those subjects labeled as applied mathematics, but at least a half of applied mathematics uses real analysis at its base, so without a good understanding of it you would not be a good applied mathematician.