Not quite -- the comment refers to the fact that R^d has the structure of a normed division algebra in dimensions 1, 2, 4, and 8. This means that you can multiply things together in a nice way when you're in one of those spaces. For R^1, this is just multiplying real numbers, for R^2 it's multiplying complex numbers, R^4 is quaternions, and R^8 is octonions. As you go up in dimensionality, you lose more and more nice properties: the quaternions are not commutative and the octonions are also not associative (which is why there's no mention of them in the blog post). The point is that in dimension 1, 2, and 4 all sorts of interesting things happen. John Baez has a paper about this: http://math.ucr.edu/home/baez/octonions/node1.html

Sort of. A Euclidean structure has “addition” but not “multiplication”. Having an additional operation that’s required to be associative, have inverses, distribute over addition, and play nicely with the norm turns out to be such a constraint that, as he mentions in the previous paragraph, the real numbers, complex numbers, and quaternions are the only such structures.

Literally speaking then, “... when R^d is an associative normed division algebra” just means “when d = 1, 2, or 4”, except of course that the idea is to use the multiplicative structure in the proof.