Derived from the theorem that a mathematician is a machine that transforms coffee into theorems.
Ok other thoughts. Go listen to some hard stuff. I loved a talk by Ron Graham about problems that would never be solved by computation. I remember feeling like I followed but all that remains is awe and images of infinite arrows.
Knowing I'll never be a mathematician but intrigued by topology, I thought about trying to write “Idiots guide to Algebraic Topology” as a way to push beyond mobius strips and klein bottles. I've made it through an intro and some insights seem sort of trivial, more along the lines of how to write down precisely an action like a twist. We know intuitively what a twist is, what a bounce is, getting it exact is hard work. But it relates to a lot of stuff we grasp as denizens of a 3D world.
Draw lots of pictures, get really good a graphing. Graham seemed to dash off quick graphs that gave quick insight, I expect he has incredible amounts of practice.
Reading some history of math has helped me.
I also think talking/arguing/chatting/ranting with peers that are very interested in topics in the general range will go along way.