Apparently this is all to do with a parcel of land near Aalborg which the Danish-American Friendship Society bought in 1912 and then donated to Kongeriget Danmark to be replanted with heather.
The one condition of this gift, to which Christian X agreed, was that this corner of the north of Denmark must henceforward celebrate US Independence Day each 4 July. So it does, and American celebrities pitch up each year to help the locals sing patriotic songs, eat hot dogs, and so on.
This piece from the LA Times is nearly thirty years old, but the custom doesn't seem to have changed since.
Yes, I know. It's a private organisation of people with ties to USA celebrating USA's independence. Gifting the land to the State of Denmark with a clause on it for its use one day every year doesn't make it an official "state" celebration.