Basically two things: 1. Complicated operations in the time domain like convolutions become simple operations in the frequency domain. 2. It is way easier to handle complex numbers and do analysis with them than with explicit frequencies to the point where some things like stability and robustness can be judged by simple geometry (e.g. are the eigenvalues within the unit circle) or the sign of the imaginary part.

EDIT: I forgot the most important simplification of operations: A derivative in the time domain, becomes a simple multiplication by s in the frequency domain. So solving Differential Equations of the system's dynamics becomes pretty easy without having to go back into the time domain explicitly.