People only stormed the Bastille to steal the gunpowder for the Weapons they stole from Hôtel des Invalides. Why is the former decided by most historians to be the start? Also, some historians consider Day of the Tiles as the start of the revolution.
The Bastille was a prison where people were sent with lettre de cachet mostly. Meaning, it was only by royal order, often without judgment.
A lot of the people there were considered enemy of the monarchy, or at least of the monarch. In fact, there was not that many prisoners inside and most were not politically significant to the revolution - nobles, clergymen,...
On the other hand it was almost mythical in its role. And the revolutionaries themselves made a huge deal of it : the king's prisoners had been freed from a (small) fort !
One year later, they commemorated the event with the "Fête de la Fédération" - it was an occasion to take oath to the new Republic and to celebrate the new national guard federations.
It became the national holiday in 1880 (almost a century later) but it is not clear if it is the Bastille day or the Fete de la federation that was celebrated at first. The current representation is mainly - only - of the Bastille Day.