Fourier transform is so fundamental to understanding the world and mastering Fourier transform (and linear algebra too) is so sooo important before entering anything "quantum". Quantum field theory (QFT), especially, is essentially a way to look at the world through its Fourier transform.

In a nutshell Fourier transform, the derivative operator, exponential function and CAUSALITY are related at a very deep level. And that's why it's such an important mathematical tool. Because it is related to causality.