From my understanding, Euler had been working on using integrals as solutions to differential equations. Laplace was extending the work. The key property of the transform is that it turns L[f'(t)] into sL[f(t)] - f(0). This can be seen by integration by parts. So once you see that, it's only a matter of time before it occurs to you (if you're as smart as Laplace) to use that property to transform differential equations into algebraic equations.

With regards to the electron, I think Ninja was saying that you can't calculate a position for an electron using an equation like that (from one of the earlier questions asked on the thread).