What is an intuitive way of explaining how the Fourier transform works?

Answer by Hadayat Seddiqi:

I realize this question is 2 years old (I remember learning originally from Mark's answer and commenting on it), but I'm surprised to have not seen this image here yet. It's very enlightening and a good supplement to the very good answers here already.

The creator makes similar animations for many Wikipedia topics, see: User:LucasVB/Gallery

What is an intuitive way of explaining how the Fourier transform works?

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