[nphrk]

This is a nice way to see how the DFT is computed, however I find the view of the FT as a change of basis as even more important - generalizes easily to other bases and and one can understand easily wavelets and their advantages. Basically, the sinusoids form a basis of the vector space of functions (every 'non-pathological' function can be written as a possibly infinite sum of them) and the numbers computed by the FT are coefficients for the respective basis vectors - the magnitude of these coefficients is interpreted as the strength of the corresponding wave in the original signal.

Another way to see the FT is as the basis where the convolution operators are diagonal - this is used in image processing, where computing the FFT of a filter + entry-wise multiplication can be much faster than running the convolution at each pixel of the input image.