After downloading Neu, you simply need to unzip the archive (your browser may have done this for you already), and move the file called "Neu" into your Applications folder.
To use Neu for the first time, double click on the Neu icon to launch it.
Enabling The Service Menus
Neu makes the "Create Document..." and "Create And Open Document..." menus available in certain places.
When Neu is running, you can find them in Neu's menubar, and also from the small menu it installs at the top-right of the menubar, or from the menu that you see which you right-click on the Neu icon in the Dock.
Neu also uses a Mac OS X facility called "Services" to add its menu commands to the Finder. This allows you to use Neu even when it isn't running.
However, Apple make users explicitly enable Services commands before they can use them (presumably to prevent applications from cluttering up the user's menus with services that they don't want).
To do this, choose "Services/Services Preferences..." from the Apple menu.
You will see a list of services (there are lots of them, supplied by many different applications). Find the two commands in the "Files and Folders" section that are called "Create Document..." and "Create And Open Document...". Tick these commands and close the System Preferences window.
You should be good to go now.
If you can't find the "Create Document..." and "Create And Open Document..." commands in the Services preferences, try logging out and back in again - this should refresh the list of commands that the Finder displays.