Part of the reason I went the direction I did was to understand how things work from the bottom up. I have learned all about circuits from the electron up.
Have you? Excuse my skepticism -- Fourier transforms show up frequently in physics, especially the statistical mechanics that underlies semiconductor theory!
Indeed, a transformation is incredibly useful when applied to repeating structures -- in this case, crystalline solids. The physics application is to integrate over a periodic structure, instead of trying to add up each and every 10^26 or so particles' contributions, surely a hopeless endeavor.

So you see terms like "phase space" or "k space", which plot wavenumber -- that is, spacial frequency (units of 1/length). The transform works just the same with respect to space, as it does to time (hertz == 1/time) -- anywhere you have a repeating structure with respect to some parameter, you can do a transform, and you'll probably find something meaningful.

Even more fundamental than that, matter itself is made of waves (quantum mechanics); FT is often used in QM solutions, though the boundary conditions are often more complex, so that an approach from differential equations is necessary. (For example, the field in a hydrogen atom gives rise to discrete energy levels; but the orbital parameters (spin and angular momentum) give rise to spherical harmonics -- the FT of waves on a spherical shell.)
Understanding wave-particle duality, is simply understanding the Fourier transform -- and once you have an intuitive understanding of wave mechanics, my friend, you can understand literally 90%, maybe 98% or more, of all phenomena in the universe, from the smallest subatomic to the largest galactic-cluster scale!
I asked my teacher and he just said so that we can analyze circuits. That didn't exactly answer my question and that is why I asked the same question here.
Speaking of analysis, and periodic structures, an interesting application of FT is the classic nerd snipe:
https://xkcd.com/356/By symmetry, the nearest-neighbor pair of points is easy to solve; but the knights-move pair shown here is surprisingly challenging. A typical solution is to consider the infinite array of nodes -- we're just doing nodal analysis like we would any other circuit -- and take the (2-dimensional) Fourier series of that entire system, with respect to position. The two points in question are the boundary conditions, and, crank crank crank, out pops a... factor of pi?

Tim