If you log in and have this checkbox ticked and you stop working at all with your pages but don't log out, this feature will be useful because next time you access the pages you won't have to log in again. It’s something like persistent connection. It works like e.g. Google account - if you check it, you should stay logged into the system. You don't need to fill your credentials anymore (until you click the 'Sign out button').
Many customers think that it provides auto-filling functionality of the username and password fields so that they wouldn’t need to enter them every time they try to login to the site. It is handled by browser itself. Please take a look at the below links for more details.