The spectral representation of a stationary process breaks it down into different sinusoidal parts 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 terms of its frequency components, we can gain insights that may not be obvious when looking at its autocovariance function. This approach is useful in various fields, like designing structures to withstand random forces or analyzing large datasets quickly using the fast Fourier transform. It's especially helpful for studying multivariate processes and can reveal important information about the underlying patterns in the data.