You yank a good few hairs (30ish preferably, WITH the root/tag attached) and send them in. I don't know exactly how they run the tests, but they know what to look for. The comprehensive agouti test tells you if the horse has any recessive 'a' and if they have any brown agouti 'At'. Anything that isn't 'a' or 'At' is bay agouti 'A'.
If you get results back for your horse and it's 'AA', 'AAt', or 'Aa' your horse is bay. 'AtAt' or 'Ata' is brown. 'aa' is no agouti. Confusing, I know.