These results are from using SNP chips. To make a SNP chip, a sample of individuals from a population (in this case, humans of European descent) are sequenced, then the sequences are compared to find SNPs (single nucleotide polymorphism: i.e. a place where some individuals have one DNA base and others have a different one.) Then some hundreds of thousands of those SNPs are selected (we want something like an even spread of SNPs over the genome, and we want to chose SNPs which have a fairly high degree of polymorphism - we'd rather something which was 50:50 rather than 99:1.) A SNP chip is designed which when exposed to DNA from an individual will say yes/no for each SNP. (Scanning the paper, I see two of the SNP chips they used were UK BiLEVE Axiom array and the UK Biobank Axiom array which have over 820,000 SNPs each.)
This has several consequences. One is that the SNP chip is of limited use for populations other than the one for which it was designed. Another is that seldom is the SNP on the chip directly related to the feature/quality (intelligence in this case) that we are trying to correlate with. Rather, the SNP which correlates positively with IQ is probably just nearby the genetic difference which matters. Because they are close, recombination (shuffling of the two genome copies you have, which happens in the production of gametes) is unlikely to separate them. Because they will occasionally get separated, the correlation of IQ with the SNP is going to be a little less strong than the correlation of IQ with the actual variant gene (allele). A SNP chip is less informative than a full genome sequence, but is much cheaper, and much easier to analyse.
A final point is that genome wide association studies like this have in the past been plagued with false positives. When there are so many variables being tested (hundreds of thousands of SNPs on the SNP chip) some will strongly associate with your measured quality (IQ) by chance. This is even more so if you use sophisticated analyses which look for combinations of SNPs as predictors. I will provisionally accept that they've accounted for this correctly, as I lack the expertise to judge for myself.
I work in a tangentially associated field (phylogenetics) so my knowledge has some professional basis, but is well short of that of an expert in the field.