>Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly.
If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof.
The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.
If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof.
The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.