The most likely explanation is that root’s .profile or .bashrc file contains or indirectly runs the command exit or logout. You can fix this by SSHing in to your droplet with a specific command and thus bypassing Bash so that you get a chance to repair the damage; e.g., ssh root@IP_ADDRESS_OF_YOUR_DROPLET vim .profile will connect to your droplet and open .profile directly without invoking Bash.

Did you find a solution?