it looks like you still have the old DLLs installed. These aren't always removed when using Unity's package management.
Make sure to delete the LeapMotion folder and the LeapMotion dlls in /Assets/Plugins/ before importing the packages. (You should be able to delete those and then reimport.) EDIT: I see now that you are using a new project. See post below.
EDIT: This isn't a DLL problem, but a prefab problem in the example scene
I'm not sure how this is supposed to be setup, but I have it working with all Dlls removed fromt he /Assets/Plugins/ (where they were by default), and the dlls imported into /Assets/LeapMotion/Core/Plugins/ and ONLY the InteractionEngine dll in the /Assets/Plugins/
As far as I know LM includes the following Dlls: