I am NOT a Mathematician, but FWIW, here are my antidotes to the crisis in meaning that you are facing:
1. You most likely wouldn't have had a problem solver's education. So the first thing to do would be do start understanding how problem solvers approach Mathematics. I suggest two books:
- The Art and Craft of Problem Solving by Paul Zeitz
- How to Solve It by George Polya
There are also others, e.g., http://www.amazon.com/Math-Olympiad-Resources/lm/1SWDJ5NA047...
Keep in mind that most proofs you see are highly polished for the sake of clear presentation. The route Mathematicians take to reach the proofs are almost always messy. And, unfortunately, they like to clear their tracks.
A great way to learn how to solve problems is to find a good problem solver and asking them to think out loud while they are solving problems. Also, try to find alternative proofs of any proof you learn.
2. Understand the history behind what you're studying. How did the ideas you're trying to learn (Abstract Algebra, Topology, Vector Analysis, etc.) come about? This will add an immense amount of meaning to your subjects.
I recommend the following books:
- "Men of Mathematics" by ET Bell. Also "The Development of Mathematics" by the same author.
- "A Concise History of Mathematics" by Dirk Struik,
- "A History of Mathematics" by Boyer
- And of course, the internet.
3. Read a few "big picture" books side-by-side. A few suggestions:
- "Foundations and Fundamental Concepts of Mathematics" by Howard Eves
- "Concepts of Modern Mathematics" by Ian Stewart
4. IMHO - and this is just my preference - study application of Mathematics to Physics and Engineering. Physics especially was an inspiration for a lot of Mathematics, and in addition, this will also let you solve concrete problems using the tools you learn.
5. During my undergraduate and graduate periods, I found several Schaum's and Dover books helpful. They're usually short and pretty cheap.
6. Initially, do not let rigor get in the way of understanding the content. Fourier, Newton, Euler, etc. weren't all that rigorous by modern standards.