I think this article is not "martingales for laypeople". It is "martingales for math nerds" where the application is solving certain types of math problems to mathematical standards of rigor. For that rigor, you do need the sigma algebras. I would say also, the target reader of the article already knows what a sigma algebra is, so the formal definition is given only for review purposes, plus to motivate something the author does about finding a disjoing minimal cover of the sample space. I haven't yet read the article carefully enough to understand why he does that.