The big thing about the Free Cities is that they had 'Imperial Immediacy'. This meant that, in theory, they were subject only to the authority of the Emperor, rather than any intermediary ecclesiastical or secular Lord. Eventually, this meant that they were--in essence--Imperial Estates, much like the other princes of the Empire. I'm not an expert in Holy Roman constitutional law so can't go into too much detail, but eventually this let them sit in and vote in the Imperial Diet, with the rights and privileges thereof. In the Diet, they sat as a third college alongside the Princes and the Electors.

Hopefully this answers what you were looking for. I'm away from most of my library at the moment, but I will post sources as soon as I can double check what says what. I believe that Peter Wilson's The Thirty Years War: Europe's Tragedy begins with a good summary of Holy Roman Imperial institutions, so I'll leave that there for now while I properly source/provide additional reading.

Edit: Source; disclaimer.

Add on question (I've asked this before but never got an answer): A Burgher was a citizen of a Free City, but how did attaining Burgher status work? As I understand it, it's possible to buy Burgher status with money, but was it also heritable? Is the son or daughter of a Burgher also a Burgher? What if they married a non-Burgher--what happens to their children then?