The colour-coded bullet points mark subspecies groups, which highlight distinct forms and their relationships.
Further explanation from the Introduction to Volume 1 of the HBW and BirdLife International Illustrated Checklist of the Birds of the World of subspecies groups and the coloured bullets:
Subspecies groups—These are informal taxonomic units used in several recent world checklists to highlight seemingly monophyletic groups of taxa (sometimes single subspecies) that at present appear to sit between the species and subspecies levels (although in some cases it seems likely that fuller scrutiny and better evidence will result in their being awarded species rank). Such groups are identified by their possession of one or a number of reasonably distinct characters and which therefore seem worthy of notice (but in most cases no attempt has been made to score these with the Tobias criteria, and no threshold number has been set for the recognition of such groups). They may already have been recognized as species in other lists or accorded a taxonomic status such as “megasubspecies” or “allospecies”, and may already possess English names (which we typically make use of). They may sometimes, however, result from our own work in applying the Tobias criteria, and for these groups we commonly supply our own English names. As a general rule, groups identified through the Tobias criteria were reasonably well marked; the weaker their distinctiveness became, the less likely they were to be separated as groups so that, if a name already existed for them, this was simply given in the Other Common Names section. Also, when an alternative name for a subspecies-group exists it is given under Other Common Names. In some cases a potential group could not be defined because one or more of the taxa involved has or have not been sufficiently studied to determine its or their affiliation.
Bullet points—In this checklist the convention is that subspecies that do not separate into groups retain a black bullet point; those that do separate into groups have bullet points coloured according to group, with the English name blocked out in the same (but more subdued) colour, always with blue for the group with the nominate subspecies, and always with the next groups coloured in the same sequence: red, green, etc.; but the nominate group need not be the first in the sequence, so bullet points may also run red, blue, green, or red, green, blue, entirely depending on the appropriate sequence of the subspecies overall. In many cases the subspecies-groups do not distort the geographical order of the subspecies themselves, but particularly complex patterns of distribution sometimes require alterations to the geographical order as a means of keeping the subspecies together in the appropriate group.
Furthermore, the same colour coding system used for the bullet points is repeated in the illustration section of the species page; the same colours are used in the headers for the figures of each subspecies, clearly marking the same groups and helping the user make clearer connections between the texts and drawings.