Mac OS X 10.4 Tiger
There is a system-wide Scripts menu that appears on the right side of the menu bar. You can turn on the Scripts menu with the AppleScript Utility in your /Applications/Utilities folder.
Place your OmniOutliner AppleScripts inside ~/Library/Scripts/Applications/OmniOutliner Pro (drop the Pro if you're using Standard), where ~ is your home folder. Then, while OmniOutliner is open, just click the Scripts menu and select a script to run it.
Earlier versions of Mac OS X
Place your OmniOutliner AppleScripts in the folder ~/Library/Application Support/OmniOutliner 3/Scripts, where ~ is your home folder. These scripts then become available in the script menu (and the toolbar). Select a script from the menu to run it.