There's an online lecture here (actually two short lectures), about how you can get from an infinite series with a general form: \( \sum_{n=0}^\infty a_n x^n \) to the Laplace transform \( \mathcal F(s) = \int_0^{\infty} f(t) e^{-st}\, dt \) with some changes so you get the continuous analog of a discrete summation. The changes are quite modest and basically about making sure everything converges. Arthur Mattuck appears to be saying the LT is mathematically inevitable, you only need a way to extend discrete sums into a continuous domain.
The series (if it has a convergence domain) represents an analytic function. This can usually be represented as the Laplace transform of something.
