Generally speaking, it is a good practice to close your web browser after accessing any website that requires a username and password. It is especially important when you are on a public computer. It ensures that the next person to use that computer won't be surfing the net as you.
When you log in to CAS, a cookie is stored in your browser that allows you to be automatically logged in to other sites. This cookie is set to expire at the end of your browser's session. So, when you close your browser, this cookie no longer exists and you can no longer be automatically be logged in to any more sites without specifying your password again.