The spectral representation of a stationary process breaks it down into different sinusoidal components with random coefficients. This helps in understanding the process better, similar to how we use Fourier analysis for regular functions. By looking at the process in the frequency domain, we can see important patterns that might not be obvious in the time domain. This is useful for things like designing structures that can withstand random forces without resonating. It's also great for analyzing complex data sets quickly using the fast Fourier transform.