That's the primary reason to use the Laplace transform, as seen in the video. A derivative x'(t) gets transformed into a product (and an initial condition), s

That's the primary reason to use the Laplace transform, as seen in the video. A derivative x'(t) gets transformed into a product (and an initial condition), s X(s) - x(0), and similar for higher derivatives, so a differential equation transforms into an algebraic equation, which can be solved by rearranging. This video assumed the initial conditions like x(0) = 0, and its notation was quite sloppy/confusing in places, as it didn't clearly distinguish the names of the two functions, x(t) and X(s).