D in this case refers to a specific set of variables that goes into brewing coffee. P(D) then refers to the probability of a given set of values for that vector of variables given all the possible values.

Don't take it too literally - P(..) here is not some well defined function, it's effectively just part of the name, as a convention for naming probabilities. I find it confusing too.

As the article points out, that set is for all intents and purposes infinite in this case, but this doesn't matter, as you can sidestep it by comparing to complementary hypotheses (which makes P(D) cancel out). This is all covered in the article.

The only maths worth reading up on to understand this article is a basic introduction to Bayes theorem - the wikipedia page is quite decent.