For example, I understand how this make sense for the Fourier transform. When we do the transform, we get 'how much of each frequency' in present in the signal. We get the value of the coefficients, such that when we multiply each coefficient with the corresponding constant frequency complex exponential and add those up, we get back our original signal. When engineers analyse the graph of amplitude squared in the frequency domain, they get the idea of how much of each frequency is present.
Is the same true for the Laplace transform as well? I know that Laplace transform is about getting "components" of a signal in terms of decaying complex exponentials, as opposed to unit magnitude complex exponentials in Fourier transform.
I haven't read much in detail about the inverse of Laplace transform. But its formula does not seem like we're summing up 'components times decaying exponential' over the entire s-plane. So what do we see when we look at, say, a colored graph in the 's-domain'? Is it really giving us the strength of the signal at each point in the s-plane? How is that true if we are not summing over the entire s-plane to get back our signal? Is this question clear?