The Remember Me checkbox only saves the User ID, and not the Password.
If you hover over the ? (question mark icon) text will display that states that it only remembers the Username and not the Password.
There is an Enhancement request for this https://blackbaudk12.ideas.aha.io/ideas/K12CO-I-830 which is requesting this should state right below the ? Icon "Will only remember username, not password for security reasons." so you do not have to hover over the ? Icon. You can Vote for the Idea as well as subscribe to updates to see about including this functionality in the possibility of a future release.