On Mac OS X, simply drag the application into your Applications folder and run it. On Windows run the installer to install Origami. Once it is installed, run it as you do any other Windows application.
The first time you run Origami, it asks you to install a companion plugin into Adobe Illustrator. Please do so. The process is explained here in detail.
New versions of Origami will require you to update the plugin. To update the plugin, follow the on-screen instructions.