The trace is about the strongest general way we have to linearly project a non-abelian situation (matrices) to an abelian situation (scalars): tr(AB)=tr(BA). By using the trace, the representation theory of non-abelian groups begins to resemble the representation theory of abelian groups, i.e. Fourier analysis. (Note though that the correspondence becomes less tight when considering triple products: tr(ABC) != tr(CBA). In particularFor related (though not identical) reasons, the theory of tensor products of representations is far richer in the nonabelian world (Littlewood-Richardson coefficients, etc.) than it is in the abelian world (convolution), and characters aren't always the best way to proceed here.)
This of course raises the question of why Fourier analysis is so miraculous, but I tend to take that as axiomatic. :-)
