Laplace transformations for dummies Is there a simple explanation of what the Laplace transformations do exactly and how they work?  Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical transformations.

I searched the site and closest to an answer was this. However, it is too complicated for me.
There are beautiful video lessons at MIT Opencourseware. I'm particularly in love with this presentation of the Laplace transform.
Think that the laplace transformation is a kind of a machine, the machine eats function of t f(t) out comes F(s). you do a transformation from time to frequency. Inside the machine you have this integral expression that you already know. it is similar when you transform from one vector space to another. for instance you go from R to R^2