Go to updatecase and make changes to your site. After clicking "Mark Live", click "Test" in the top right to apply changes to Testing
Select all the files within the live site directory then click "Move selected" button
Navigate to your "backups" directory and create a folder. Naming it the current date (e.g. 2015-05-05) will allow you to find it more easily if something goes wrong.
Go into the newly created folder and click the "Select" button (this will move the current live website files into this backup directory)
Navigate to the live directory and click "Select" button to move Testing files to Live
Below are many other instructions that show you how to use your UpdateCase application