He also refers to a lot of background knowledge I just don't have, e.g. I don't know what I'm supposed to take away from him calling something Calvinist.

Could you please clarify if you are unsure what he means when he calls a position Calvinist (presumably Crypt-Calivinist or something like that) or are you just unsure what Calvinism is?

The short and sufficient answer to the second is that this is a designation for a bunch of Protestant Christians who historically took themselves very seriously and have a reputation for being dour. Take special note of the Five Points of Calvinism.

The short and insufficient answer to the first is people who have ethical, political and philosophical ideas that can't be justified by their declared systems of ethics but can be perfectly well explained if you note the memeplexes in their heads are descendent of highbrow American Protestantism of the previous centuries. He goes into several things he considers indications of this and points out they dislike this explanation very much and want to believe their positions are the result of pure reason or Whiggish notions of history inching towards a universal "true human morality".