It's a bit late here, so maybe not four paragraphs, but basically what Laplace transforms do is, you feed it some sort of differential equation or system of differential equations in a variable
t and it transforms these into some other space where the variable you work with is
s, you then solve this in the
s-plane so to speak, because it's
much easier to solve things in the
s-plane (sometimes) and then you convert it back to the
t-plane. So it's sorta like integration substitutions in the sense, you substitute something (laplace transform), the integration is easy (solving the problem in terms of s), you integrate it in the new variable (s plane) and then transform it back to the t-plane (back-sub) where the brackets are used for the analogous action.
A good way of explaining this in a "analogy" way is:
But I'm afraid that the above only tends to make sense once you've studied a teensy bit about Laplace transform and know how to do the mechanics behind them, the above then explains in a certain fashion why the mechanics works the way it does and what motivates it.
If you're interested in this sorta stuff, there is an excellent introductory lecture
here which has garnered international acclaim. :-)