Okay so I'm going to give a very brief overview of the conflict as it is very complex and complicated. So there are two sides, Bismarck and the Austrians. Bismarck wanted to annex both Schleswig and Holstein to Prussia, while the Austrians wanted them to be an independent state under a German dynasty. The reason they were able to cooperate was because they both wanted the territory out of Denmark's control, but once it was the relationship broke down. Now since Austria and Prussia couldn't agree on an actual way to divide the territory, they agreed to both occupy it. Austria got Holstein (I'm not sure why, sorry.) and Prussia got Schelswig. Anyways Austria didn't want to hold onto the territory because it was far away and expensive. So what happened was Austria withdrew giving their part of the territory to the German confederation, which was a very loose body of all the German states. Prussia then marched its troops into Holstein, the German confederation condemned them, Prussia essentially said they don't care what anyone else says. And the war started five days later.

That's the really simplistic version, let me know if that clears it up for you.