It means rain is on the way. This saying seems to be more widespread around the country nowadays, but many - including Eric Partridge in his 'Dictionary of Catch Phrases' - believe it started life in the East Midlands. The best explanation I've ever heard as to its origin is the one that suggests the 'Bill', 'Will' or 'William' in the phrase refers to William (Will/Bill) Shakespeare. He was born in Stratford-on-Avon, to the west of the Midlands and - if the sky is dark and threatening over that area (ie over Bill's mother's) - it means rain is probably headed for the east of the Midlands region, because the prevailing winds are likely to push the wet weather in that direction.