In order to log in without an internet connection, the user must have logged in to the Chromebook at least once when it was connected to the Internet. When a new user logs in to a Chromebook, a profile for that user is built in the background and is what allows the user to login when the Internet connection is removed.
Step 1: From the login screen, click the "sign in as an existing user" link:
Step 2: Enter your full email address and log in as you would normally.
You should now be logged in to your local Chromebook profile.
(We apologize for the poor quality of these screenshots, but it is very difficult to take a good picture of a computer screen!)
For more information, please visit:
Chromebook Help - Use Your Chromebook Offline