This option allows you to stay signed in until you specifically ask to be logged out. If this option is unselected then you will be logged out whenever you close off the internet or leave the site. Selecting it prevents this from happening and keeps you signed in whenever you go back to the site on that computer.
We store what's called a cookie on your computer. Whenever you come back the site recognises this and logs you in with your password.
NB: If you're on a shared computer or a public computer then we advise you not to select this or to ensure that you select log out to ensure the next person who uses the PC doesn't log in to your account.