The kind of understanding and appreciation you're talking about, where you "see" the math, can be hard to get to. For me I never found it until I started doing

The kind of understanding and appreciation you're talking about, where you "see" the math, can be hard to get to. For me I never found it until I started doing math in applied contexts such as physics or computer science. My "eureka" moment was in a physics course where we were looking at the derivation of the differential equation for a damped harmonic oscillator. One moment the equation was something I had seen many times and knew how to solve problems with; the next, something "clicked" and I understood it, where the terms came from, why it all made sense, in a way that's hard to describe. Visualizing the oscillator and knowing the equation were the same thing in a weird way. After that, and a couple more similar moments, I found it much much easier to understand and appreciate not only applied math but also purer concepts.