The Nash Equilibrium is, at its essence, an application of a fixed point theorem. If you are interested in building the mathematical background to understand Nash Equilibria, you must simply learn the math required to understand fixed point theorems. There is no predefined path for learning about fixed point theorems. One usually learns them in a topology course, but they are neither the focus nor the goal of the course, so you will need to form your own path out of the requirements.
If you have a strong idea about proofs, and a high mathematical maturity, then you could probably work through the standard text in Topology by Munkres right off the bat. If you haven't worked with proofs or elementary set theory in a while, you should get reacquainted with proofs, and I can't imagine a better way to do both than work through Halmos's Naive Set Theory.
In general, check out the topology books listed as an answer to this question. Note that one is free; I would recommend starting with that before purchasing Munkres. If you don't like it, at least it will give you an idea of what subjects you need to know in order to approach Munkres.
After you have understood the ideas of Topology you can move on to a book on game theory which covers the Nash Equilibrium proof. Here are three standards:
Playing for Real: A Text on Game Theory by Ken Binmore
This is a text for advanced undergraduates. Chapter 8 covers what you want to know, however it is limited to the two-dimensional case (which is much simpler to prove, however the ideas are the same in higher dimensions). If this is the book you wanted to go with, I would suggest just buying it and foregoing all the Topology stuff. By omitting the higher-dimensional cases, it is able to stay remarkably self contained.
A Course in Game Theory by Osborne and Rubinstein
This is a graduate level text, and is probably more along the lines of your interest. The only deficiency that I can see is that it does not include a proof of the Kakutani fixed-point theorem! It simply mentions it and uses the consequences to conclude the existence of Nash Equilibria. You still get a good idea of why it happens (the exercises force you to play with the restrictions, and this gives you insight into why it is true - you definitely need the Topology books first for this one though), but if you want a proof then you will not find it here. This book is a standard though, and inexpensive, so it is worth getting so that you can read it along with...
Fixed Point Theorems with Applications to Economics and Game Theory by Kim C. Border
This is really just a collection of notes, but a very good one. It is a standard reference for this subject, and does include a proof of the Kakutani fixed-point theorem. This is not really a good book for self study, however; it is intended as a reference. I would suggest getting this book as a supplement to Osborne and Rubinstein.