I've started learning about spectrum analysis a long time ago at university, but failed to gain a true insight in how it works under the hood. For me, it was just some magical math formulas I had to memorize and know about. I knew the theory but it didn't really "click".
A few years ago I wanted to do some audio programming and play around with analyzing sound. That's how I got interested into building an intuition on how the Fourier transform "actually" works.
Source: Demystifying Fourier analysis, an article by Davorin Šego.