You can mark a task as complete by opening the Outlook task, and selecting Completed for the status, or clicking on the "Mark Complete" ribbon button in the task. The app will mark the task as complete, and it will also complete all the task's children.
This screen shot shows the effect of completing "Project 1": Project 1 and its children are marked complete, and the status of linked Outlook tasks is set to Completed.
The app will show tasks that you completed today. Tasks completed before today are not shown in the list.