If you made your purchase though the iTunes Store, your updates can be managed automatically through iTunes. If you purchased OmniOutliner from The Omni Group's online store, please read on.
OmniOutliner can automatically use your internet connection (when active) to check for new and updated versions of itself. If the Check for Updates checkbox is selected, you will be notified when there is a new version of OmniOutliner ready for you to enjoy.