You're probably just missing updated graphics card drivers (most bundled drivers are missing OpenGL support which is required for Hedgewars to run).

The (right now) most up to date driver should be the one available for download here: http://www.nvidia.com/object/notebook-win7-winvista-64bit-258.96-whql-driver.html

The installation should be quite self-explanatory/automated. All you have to do is to download the file and run it.

Also try to disable fullscreen in the game settings, as you might have to grant access in the Windows Firewall first (which could block the game from showing anything). Enbable it again once you've got it running.

If the above steps don't work, provide us the contents of the file "debug0.txt" that can be found in "C:\Users\[Your Windows Account]\.hedgewars".