First off, nobody in France calls it that. It's the Fourteenth of July, just like the Fourth of July in the U.S.
Second, the 14th of July isn't an Independence Day. It's formally called, La Fête Nationale (The National Celebration). It marks the anniversary of the storming of the Bastille prison as part of one of the earliest attempts by the French people to dump the monarchy and the feudal system.
Third, that whole thing about storming the Bastille to liberate political prisoners isn't exactly true. Only seven people were held at the time: four crooks, two "lunatics" and according to Wikipedia, one "deviant" aristocrat.
Fourth, the real reasons the Bastille tower and prison mattered in 1789 was that it was a symbol of the monarchy's abuses. And more importantly, a stockpile of weapons and ammunition. The angry crowd ripped it down stone by stone. And being thrifty, later sold them for souvenirs, some fashioned into replicas of the fortress.
Fifth, it didn't entirely work. The brief republic of the French Revolution and the Declaration of the Rights of Man and of the Citizen, and the abolition of feudalism, not to mention the Reign of Terror, gave way to the imperial reign of Napoleon, more monarchies, brief attempts at republic, until the thing finally put down deeper roots with the Third Republic in 1870.
To get a taste of Bastille Day in France, from the parties to the parade, and the political context, check out my video.