I'm not sure this is what you're looking for, but the Feynman lectures give the basic idea in an elementary way: https://www.feynmanlectures.caltech.edu/I_23.html

> This idea of using exponentials in linear differential equations is almost as great as the invention of logarithms, in which multiplication is replaced by addition. Here differentiation is replaced by multiplication. . . . See how simple it is! Differential equations are immediately converted, by sight, into mere algebraic equations