There’s an XKCD comic which explains the problem. Unfortunately, that comic is too big to post here. Briefly, a p-value of 0.1 says (roughly) that there’s a 10% chance (0.1) of the observed result being as extreme1 as it is simply due to chance (sampling variation from a population), assuming the null hypothesis is true.
Often, 5% is more or less arbitrarily chosen as “cut-off”: results with p-values below that (which occur by chance 1 time out of 20) are called significant – results above are insignificant.
However, this directly implies that when doing multiple studies, say 20, there’s a very high chance of getting a “significant” result by pure chance, without there being a real effect.
GWAS studies essentially perform experiments on many (hundreds or thousands) of factors simultaneously. Therefore, reporting the pure p-values would be very misleading since there would be a lot of spurious results with “significant” p-values. The probability of getting one or more spurious hits is called the familywise error rate (FWER).
The FWER can be reduced by adjusting the p values, for instance by performing the Bonferroni correction. The resulting p-values are quite a bit higher, which reflects the reduced certainty that the result is due to chance, i.e. the reduced significance.
It doesn’t generally make sense to report the unadjusted p-value. It was probably reported here because correcting for FWER is quite conservative (i.e. it trades a low false-positive rate for an increased false-negative rate) and the authors feared that the adjusted p-value looked less impressive than it should. This involves a fair bit of interpretation.
1 extreme = different from the expected result under the null hypothesis