I took an introduction to ODEs course this past spring semester. It always bothered me where this thing came from. I did a little bit of research and found a video of a professor explaining how it is the continuous analog of an infinite sum. He did a little bit of a derivation using that analogy. That's fine and dandy, but its there a rigorous way to derive this integral expression? What does it mean and what does it tell you about the function you apply it to (i.e., what is the relationship of a function and its Laplace Transform)?