This is something about politics in England.It would be better if you gave us the exact text (where it is taken from ) and the context. Chequers is the residence of the Prime Minister of the UK, and a bluebottle is a kind of fly,so this relates to the idea of a "fly on a wall",someone who can listen into a private conversation without being noticed or seen.The franchise are the people who are entitled to vote in an election.SO it is referring to listening to a private conversation at the Prime Ministers' private residence.

The usual saying is "I'd love to be a fly on the wall," meaning "I'd love to hear what actually goes on at a secret meeting." Because "I'd love to be a fly on the wall" is such a stock expression, the writer decided to add just a little dash of originality to it by using the word "bluebottle" (a specific species of fly) instead of just saying "fly."