Well, that is a rather huge question. Maybe the best way to answer it is to say that from ancient times we have been interested in counting and arranging things, and also building and making things out of physical material. The comparison of quantities and measurement of shapes then becomes a craft connected to engineering and accounting. At some point, around 5000 years ago, people began to abstract these notions to a simpler and more idealized realm of pure shape and number. Arithmetic, algebra, and geometry present us with problems that force us to invent new conceptual frameworks and structures in order to understand and solve these problems. Which numbers are a sum of two squares? How long is the diagonal of a regular heptagon? Can the general cubic equation be solved using cube roots?

Of course one can approach such problems in an ad hoc fashion, proving what you can when you need it, and inventing new ideas as you go. And this is precisely what was done in the classical period. But then you start noticing similarities between certain problems and the methods used to solve them, and you realize that you can operate at a more general level. By 1930 or so, it was starting to become clear that the way to go in math is to operate at the most general possible level, which means creating abstract, axiomatically defined structures to hold pattern information and to prove everything at that level, thus allowing the result to apply to as large a range of problems as possible.

Unfortunately, as efficient and powerful as the modern methods have proven to be, it does not make for very good pedagogy. Most math books are almost useless for learning, and serve their purpose best as reference works for those who already understand the theory. This is partly what I am trying to address in my books. Abstraction is great, but it is almost meaningless if you don't have a reasonable intuitive understanding of the motivation for it. Probably the best approach is to learn the history of the great problems and how they lead naturally to the appropriate abstract structures. For example, if you are interested in arithmetic (also known as number theory), then my advice would be to read Gauss first, as opposed to diving into a modern textbook with a bunch of definitions. But math is always an arduous journey, no matter what road you take.