Interpretation of Laplace transformed function and Laplace vs Fourier
The Laplace transform has some nice properties that help to get more insight into the behavior of linear systems.

A very nice property is that the Laplace transform evaluated along the jw-axis is equivalent to the Fourier transform, which is less abstract and easier to understand. This of course brings us back to question, why we didn't use the Fourier transform in the first place.

The answer is quite simple. We often have to analyze and work with systems that are unstable or where we want to determine whether they are stable or not. In such cases the Fourier transform fails (=does not exist). This is the reason the Laplace transform was introduced. It includes an additional term that helps the integral to converge and therefore the Laplace transform can be applied to a broader range of problems and applications.