From the solution method of second degree differential equations we know that a system can be modelled as $ce^{(\lambda + \omega i)t}$. The complex constant contains information about the decay rate (real component) and the natural frequency (imaginary component) of the system.
Looking at the definition of Laplace Transform, I see:
$$\int_0^\infty e^{-st} f(t) dt$$
This integral clearly extracts knowledge on this complex constant, hence it allows us to analyze the system in more general terms. We pass from the time domain to the frequency domain.
But how does it happen? In the definition above I sense some kind of "squeeze" (where $s$ is also a complex number) but I am unable to understand it intuitively. If I can get to understand it, this would vehemently aid my study of the topic. Thanks.