The following steps have been tested using Mac OS X Mountain Lion (10.8) and Snow Leopard (10.6.8). If you have comments on the download they are welcome on the Talk:Macintosh discussion page.
Visit the Freeplane Project page at SourceForge, and click on the green "Download" button to get the latest version. As of this writing the current version is freeplane_app-1.2.18.dmg
The .dmg file will download to your Desktop or Download folder, or wherever you normally store downloaded files.
If it takes more than 30 seconds for the download to begin, you may need to click on the link at the top of the Sourceforge page which says "Please use this direct link".
When the download is complete, find the downloaded .dmg file and double-click it. This will open a new Finder window called Freeplane with a Freeplane bumblebee icon inside. Drag the Freeplane icon into your Applications folder. You can also drag the Freeplane icon into your Dock so you have easy access to it. You may now delete the downloaded .dmg file.
Click on the Freeplane icon in your Dock or double-click on the Freeplane icon in Applications. Click “Open” when prompted. Voilà.
When you open Freeplane you will be greeted with a Freeplane Applications window which gives you an idea of ways that Freeplane can be used. Click on any of the elements you see to learn how to create them and edit them.
If you’re working on a few maps together the file names will all be visible at the bottom of the Freemind window.
To exit Freeplane, choose File > Exit. Alternatively you can use the shortcut: Command+Q.
If you have feature and enhancement suggestions for Freeplane then add them to the Talk:Macintosh discussion page.