This is because Inkscape used to run on Apple's X11 window management system. However, Apple has dropped support for this in OS X 10.8 Mountain Lion.
Apple has open-sourced X11 according to this support article. It is now called XQuartz. Installing and setting up Inkscape is very easy.
- Go to this website and install XQuartz. Open the folder in the .dmg and drag the app to your Applications folder.
- Reboot your computer or log out and log in or apps that work with XQuartz, like Inkscape, will not run. This is important.
- Normally install Inkscape from here.
- Start Inkscape from Launchpad.
- It should ask you to browse to the XQuartz app in a window that comes up. If you can't find it, browse folders and find it in the Applications folder.
Once you do this, Inkscape should run normally. If you're looking to install the open-source photo editor GIMP on Mountain Lion, you will need to have XQuartz for that as well.