CCC's main window is divided into three main sections. At the top is a toolbar, and beneath that there is a split pane divided vertically. The view on the left is called the sidebar, the view on the right holds the task configuration view. When you open CCC for the first time, the sidebar is hidden. The sidebar is automatically revealed when you add a new task.
CCC's Disk Center shows general volume information for each locally-attached volume mounted on your Mac, as well as read and write rate and error statistics for those volumes. Select a volume in CCC's sidebar (click "Show Sidebar" in the toolbar if it is hidden) to view that volume in the Disk Center. CCC also shows any backup tasks that are associated with the selected volume. Simply click the Clone button to run a selected task, or double-click the task to edit the task.
To find out when a backup task last ran, click on the Carbon Copy Cloner icon in your menubar, then move the mouse over the submenu of the task you would like to inspect. The submenu for each task will indicate when the task last ran, the status of that last run, and when the task is scheduled to run next (if applicable). Select Show task history... to open CCC and view more detail about the previous times that that task has run.