How challenging did you find Real Analysis before the penny dropped? I've been following 'Understanding Analysis by Stephen Abbot' and have been struggling with exercise questions even after an in depth read of the chapters. I have read numerous statements by authors that 'if you can't do the exercises, then you haven't grasped the material properly', but I feel like I have grasped the concepts. I've read many 'how to' books to finally understand higher level math and persevered with a 'never giving up' attitude, but most exercises really throw me off. Even those in the first chapter which are meant to be on basic preliminary material. 
Is this just the nature of studying Real Analysis? I'll be starting Abstract Algebra soon and have been advised to purchase 'J.B Fraleighs introduction'. Do other higher maths  subjects cause the same anxieties as Real Analysis? 
 A: A few bits of advice...
First, draw pictures. Specifically, draw graphs of real-valued functions, and see what various theorems are telling you about these pictures. Draw pictures to help you solve problems, too. Pictures don’t constitute a solution or a proof, but they can often guide you towards the right line of reasoning. 
Second, play with lots of examples. Suppose you’re studying a theorem that has several hypotheses. Think of examples where the hypotheses are satisfied and where they are not. If one of the hypotheses is not satisfied, maybe the theorem is no longer true. Think of examples that illustrate this.
Third, don’t expect things to be easy. Mathematics texts (and many teachers) have a nasty habit of showing you only the pretty final results, without showing you all the false starts and roadblocks and mess that preceded them. You won’t be able to write textbook mathematics on your first try. The first few attempts will probably be convoluted and ugly. Once you have a solution that works, then you can tidy it up and make it look as pretty as the stuff in textbooks. 
A: I would say that the ability to do the exercises is an indication that you have (or are currently) mastering the concepts, but that simply grasping them is a much lower bar. 
My personal advice is the following: find some classmates and take turns teaching each other the material. Literally. Go find a study room or somewhere you can stand in front of a board and teach each other what you learned in class. Reprove theorems. Re-examine examples. Re-state definitions (from memory if you can). There is no better way to learn than to teach.
If that doesn't help, then my next bit of advice is to practice 'till you puke. Another answer stated something about easier exercises to cut your teeth; this is an amazing bit of advice. But don't feel compelled to get back up to the hard problems fast! Keep practicing everything, because you never know when it will all click, and what will be the cause of said click.
A: There is already really good advice here; just wanted to add some stuff which I think hasn't been mentioned yet.
When you read a theorem, avoid reading the proof right away and try for a few minutes to see if you can prove it youself. It's totally fine if you cannot prove it; the main point of this exercise is to help you understand what the theorem is really saying and to make it more memorable. If you want to do more, you can do stuff such as cover the proof with a blank paper and progressively reveal it, while trying to guess the next line or the rest of the proof.
Also, you may want to do this: 5 minutes after reading the proof, try to write it up youself (or at least a briefer summary of it); this can be surprisingly difficult. If you cannot do it, read the proof again and repeat.
Similar thing for the exercises; if you cannot solve an exercise, look at the solution and try to understand it. Once you feel that you understand it, try to write up the proof yourself; and if you cannnot, read again and repeat.
A: I have two insights I have gleaned from my personal struggle with similar difficulties:
First, if you are having trouble with exercises, find easier exercises on which to cut your teeth.  There are always easier exercises.  The point is to jump in at a place where you feel comfortable working so you can grow your skills naturally.  You may have to switch books to do it, but it can be done.  Skill grows inevitably with practice, but I think  you have to practice with what you can do.  Sometimes baby steps are the wisest way to go.  Just my thoughts . . . 
Second, in my experience as a student, teacher, and researcher, some people are just naturally better at some subjects than they are at others.  I have seen people who could breeze through graduate algebra courses struggle to barely pass analysis or differential equations, and vice versa; and I think putting time in on what you have a natural knack for may contribute to your success in subjects which you find intrinsically more difficult.
I hope these words are helpful.  You are not alone.
