It was a USO club - run by the United Service Organization specifically for American soldiers. I did some digging and did not find a ton of detail, but here are some tidbits that you may find interesting:

• Django Reinhardt (a very famous Jazz guitarist) worked for the USO in 1947, and played the club for four nights sometime in July of that year. Michael Dregni's Django: The Life and Music of a Gypsy Legend describes it as a place where "American GIs filled the hall, eager for a good time with the war over and nothing to do but drink German beer and listen to Jazz."
• Your Grandpa is apparently not the only one to have found love there: there were several news reports in 1947/8 that a Chicago man named Ray Mack (who had apparently worked as master of ceremonies at the club) was marrying one of the club's dancers, Kay Trevil (who happened to have won "Miss Paris" in 1947). According to the Chicago Tribune, she accepted his proposal (via telegram!) in October of 1947, and historical images has a photo of them on her arrival in 1948 online.
• In a similar vein, I found an article in the Annapolis Capital about a local dance instructor named Wally Dashiell, who was born in Breslau and met her husband at the Stardust Club in 1944 (again, while she was working there as a dancer). They married, she moved to the US, and she opened a dance school which she operated for many years.
• A New York Times article from 1946 ("GI Night Life Sampled," July5, 1946), describes the club as "the biggest GI night spot in Europe".

Edit: Found a bit more:

An interview with the club's manager on the Chicago Daily News wire, dated June 1949, is titled "Want to Run a Nightclub? Join the Army!" and describes the Stardust as "a ga-ga joint for privates and corporals only." According to this article, "When Gen. Eisenhower bossed Deutschland he insisted that the bottom rank GI's have a swinging saloon all their own - unpestered by supervising officers, or, for that matter, sergeants," and the Stardust was created to fill that role. It was apparently run by-and-for enlisted men, and was managed by a corporal. The reporter visited the club and described it as follows:

GI's were skidding around the wax with frauleins. I only saw two American girls - WACS. 'We have an Allied floor show,' said Pitch [the manager], 'without Germans. It cost $89.26 per night through the army's booking office. The boys are sick of German shows - they're mainly acrobats tossing each other around.'

The same article goes on to describe how it was a private club, where officers, uninvited sergeants, unescorted women and male germans were not allowed - so basically just GI's, corporals, and their dates. Their top selling drinks were Tom Collins' and rum and cokes (both sold for 35 cents each).

Another article from the Minneapolis Star (dated December 1946) states that the club was about a year old at that point, and already "one of the best known army installations in Europe," having been founded by the Third Army "with the idea of keeping soldiers off the streets at night." Apparently single German girls with "social passes" were admitted at that point (NB, this was probably a means of keeping prostitutes out - the passes were only issued to girls of "good character.") Contrary to the Chicago Daily article, this one states that the principal founder of the club was Lt. General Geoffrey Keyes, of the Third Army - which had its headquarters in Heidelberg. This article says that about "90 percent" of the women who attended were German, but it also notes that it was kind of hard to get drunk there - most of the evening was taken up by a floor show (during which alcohol wasn't served), and you could get kicked out for being intoxicated.