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.
Restoring individual items from your backup volume can be done in the Finder via drag and drop – simply find that item on the backup disk, then drag it back to your startup disk. If you're restoring more than a handful of items to your startup disk, configure a folder-to-folder task to restore specific folders from your backup disk to a specific folder on your startup disk.
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.