Get to know the Danes
Why locals love it
Or the history of Denmark, at least. The National Museum features a huge exhibition of stuff that came from around Denmark and the world. Mostly Denmark. It's a tourist thing, for sure, but if you have more than a passing interest in history and archeology, it's worth the visit.
Why you should visit it
The exhibitions are quite good and comprehensive, if you're into cultural history and such. Changing exhibitions about other cultures keep it fresh. Their collection spans from ancient times to modern history, so there's a bit of everything.
Special tip
It's for free, so even if you don't like it, at least you didn't spend anything on it.