The LCD is powered directly by the battery. It works independently from the controller, but is able to communicate with a controller provided that both have compatible software and the correct data protocols.
You can switch on the LCD by connecting the red and black wires to the battery. It will then light up as normal and display the battery voltage correctly.
When it's switched on, it also switches on a third wire (normally blue or orange) that provides the battery power for the controller. Note that the power for any controller is actually independent from the power to the motor. This third wire can then be used as the ignition wire for the controller so that you will be able to switch the controller on and off from the LCD, which is much more convenient than using a separate high power battery switch.
The other two wires to the LCD are for data (Tx and Rx), where the controller sends information about speed and power, and the LCD sens information about PAS levels and speed limits. You can't use them with a Lyen controller, so none of the other LCD functions will work.
The LCD SW900 is made by King Meter. It's compatible with LSW controllers and some Ananda ones, but I don't know which others.