H is the set of all possible Diplotypes that are consistent with genotype data
Disclaimer: Like the OP, I know very little about genetics and I suppose other people in the site can give better answers than me. Anyway, since the question has been unanswered for months I'm posting my answer. Hopefully someone will improve it.

Just beware that "possible" in this context doesn't mean all haplotypes that we can imagine. As the paper says, "$\mathcal{H}_G$ denote the set of all possible diplotypes that are consistent with the genotype data $G$" and that is an additional restriction that allows just some diplotypes - the seven ones listed, not the 32 we can imagine.

Furthermore, please notice that sum of frequencies in the table equals 1, therefore there can be no more haplotypes in the set.

However, it would help if you linked the source where you got those data and quotes.