I didn't want to introduce the notion of infinity because there are actual "infinite numbers" on the surreal number line. I've kind of tried to have both the simplicity of set-theoretic definition and the intuition of the number line, and slightly bungled the exposition. I hope the newly added diagram helps.

My favorite intro to surreal numbers is https://www.infinitelymore.xyz/p/surreal-numbers, but it is behind a registration wall.