The "magic" of the Fourier transform and DFT is entirely contained in this proof, IMO. I think a lot of people find the Fourier transform extra-amazing because

The "magic" of the Fourier transform and DFT is entirely contained in this proof, IMO. I think a lot of people find the Fourier transform extra-amazing because it is their first encounter with basis change and infinite dimensional vector spaces. The FFT is still magic though :)