Several authors I can recommend who write good expository books on mathematics to explain the why are
Timothy Gowers has a book about mathematics in the Very Short Introduction series
that should be helpful to you.
On the particular subject you are studying right now, see if Hubbard and Hubbard's book Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
helps your understanding.
Mathematics: A Very Short Introduction (Tim Gowers): http://www.amazon.com/Mathematics-Short-Introduction-Timothy...
The Language of Mathematics: Making the Invisible Visible (Keith Devlin): http://www.amazon.com/Language-Mathematics-Making-Invisible-...
What is Mathematics?: An Elementary Approach to Ideas and Methods (Richard Courant): http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...
They give brief but interesting glimpses into several areas of higher math, so they could help you decide what you want to pursue in more depth.
I'll recommend a couple of books from that thread:
I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.
Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.
Get that right away.
Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).
Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.
(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)
(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)
Concepts of Modern Mathematics by Ian Stewart
Numbers and Geometry by John Stillwell.
The Pleasures of Counting by T. W. Körner
Mathematics: A Very Short Introduction by Timothy Gowers
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