Basically the assumption is that the signal x(t) is equal to 0 for all t < 0 and that the integral converges. And what is a bit counter-intuitive: Laplace transformations can be regarded as generalizations of Fourier transformations, since the variable s is not only imaginary but fully complex. But yeah... I would have to brush up on it again, before explaining it as well. It's... been a while.

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.

I was at the intersection between mechanical and electrical engineering as well as computer science. And worked in/with (electric) mobility, agriculture, medical/rehabilitation tech., solar energy, energy grid, construction and building tech. As well as some very limited stuff with economics. And I intentionally chose my study courses to be able to work in multiple areas and inter-disciplinary. My latest work is more on the business and management side of things and less technical, though.

What are you studying and what direction are you hoping to head in?