Shorter and less accurate than the Wikipedia page: A SNP is a single nucleotide polymorphism, a single letter variation in a genome. This study measured about half a million SNPs for each person. A statistical test is then performed to see how well each SNP site predicts the trait, generating a p-value. If a single such test were being performed, then typical "significant" p-value levels would be 0.05 or 0.01 or 0.001, these are arbitrary but generally accepted. For data from SNPs unassocisted with the trait, p-values come randomly and uniformlay from the range 0 to 1. So with a hundred SNPs unassocisted with the trait, a person would expect about one p-value at <= 0.01. There are many ways to correct for these multiple hypothesis tests. For GWAS, the generally accepted significance levels are 5e-8, which under the rubric of the fancily named but simple Bonferroni correction, would be equivalent to a 0.01 to 0.05 p-value from a single test. These two reported SNPs, when correcting for multiple testing, don't meet the standard definition of significant.
They found "support" for associations between homosexuality and some specific genetic code positions, but not strong enough evidence to be very sure of anything ("significance").