I think the key is realizing that math is about manipulating a mental model of an idea. The wrong model can make certain tasks impossible -- for example, thinking that numbers must be 1-dimensional makes complex numbers a paradox. But when you see that they could represent items in 2 dimensions, you see how complex numbers represent a rotation. And from that, numbers can have arbitrary dimensions (vectors), and so on.

Whenever I study a new concept, I try to find the mental model behind it. For a book recommendation, I'm going through Visual Complex Analysis (http://www.usfca.edu/vca/) and find it extremely well written, with an intuitive approach.