Mathematically, the integral of the product of two functions is an inner product and the complex exponentials are a convenient set of orthogonal basis functions for an arbitrary function space. That's Fourier's theorem. We can define a vector space containing all such possible sampled signals. But a projection onto these basis vectors yields little insight.
Complex exponentials are sinusoids.
We hear sinusoids - or, at least, we can detect the presence or absence of sinusoids at various frequencies. Thus, such a transform is musically relevant. There are lots of other good reasons why we might like such a transform. Most significantly, the operation of filtering is very simple once we are in the frequency domain - in order to change the amount of a certain frequency which is in a signal, we just multiply that coefficient by a constant. The DFT revealed The transform I've described is very close to the Discrete Fourier Transform.