How do you go from these definitions of ordinal arithmetic by transfinite recursion to mechanical rules for arithmetic on the cantor normal form / deriving the needed algebraic identities? It seems very non obvious to me, although certainly must be very obvious to many. I would like a reference if you have one.

I think a deficiency of my post is that I didn't go into much of what the ordinals are, but I think a strength is that it is all quite mechanical.