The short explanation of the DFT that I like the most is that it projects the signal vector on to the vector room with the exponential functions as basis vectors. Then you can see how much of the signal that each basis vector, corresponding to a frequency, can explain.
It's intuitive to see if you start with a 2D vector space (the regular euclidian plane) and a 2D vector, and then you can expand the definition of vector spaces to include orthogonal functions as bases.
The w Fourier coefficient F(w) is the dot product of f with an exponential function, `e_w • f`, and is in that sense a projection. The inverse Fourier transform writes the original function as a sum of the projected components: `f = sum_w (e_w • f) e_w = sum_w F(w) e_w`. This is exactly how writing an "arrow" style 2- or 3-D vector as a sum of orthogonal projections works.
It's intuitive to see if you start with a 2D vector space (the regular euclidian plane) and a 2D vector, and then you can expand the definition of vector spaces to include orthogonal functions as bases.