Laplace Transformation is a useful tool in problem solving. From the probabilistic theory perspective, Moment generating function is nothing but a linear combination of two Laplace transforms. In fact moment generating function has a more direct relationship with Laplace-Stieltjes transform. Most of the calculations involving MGF and convolutions become easy with the use of Laplace Transform. I think one of the most powerful ways to use Laplace transforms is solving Partial Differential equations. Having the basic Laplace transforms and Inverse Laplace transforms at your fingertips is good for solving Toy PDEs. But in real life, most of the Inverse Laplace transformations have no closed form solutions. They have to be solved numerically. Coding up an Inverse Laplace transform in whatever language you are comfortable is a nice learning experience. My Laplace transform fundas were rusty, so decided to go over this book quickly. The author P.P.G Dyke, being an applied mathematician, makes the book interesting by giving a range of problem domains where Laplace and Fourier transforms can be used. This book is ideal for those who want a quick recap of Laplace and Fourier transforms.
Inverting a Laplace transform has some bit of math of behind it. Here is a paper by Joseph Abate and Ward Whitt that gives the details behind numerically inverting Laplace transform. The nice thing about the paper is that the pseudo code for the algorithm is given so that one can go ahead, code up the algo and subsequently write at least a dozen unit test cases for each of well known closed form Inverse Laplace transforms.
Use of Fourier and Laplace transform and such analytical methods have been surpassed by computers that solve a PDE using numerical methods. However analytical methods give the intuition behind the solution that is not so obvious from the numerical solution.