What does the phrase "electrically controlled and pneumatically operated" mean?
It means that an electrical switching signal is used to drive the pneumatic or air valves in the air conditioning system.

Think about it like this: We want to use our garden hose to push open a cat flap but we want electricity to control the pressure in the hose. We could apply an electrical signal to a motor which opens the tap and lets water come out. This water pressure could then be used to control how wide the cat flap opens. The only difference in your airbus situation is that it isn't water (which would be hydraulics), it's air that comes out of the tap (Pneumatic means air). So this system would be called an electrically controlled, hydraulically operated system. Substitute the water in the hose for compressed air and you have your answer.

The way electrically operated valves work is generally based on the principles of electromagnetic force by utilising  some sort of coil. This setup turns a piece of wire into an electromagnet, which is attached to or interacts with the movable part of the valve. This then ports air pressure to one side or the other, to balance open the valve to a set amount in proportion to the original electrical signal.

So we use electricity >>> this changes the balance of air >>> the air opens or closes the valve.

The valve can be fully open, fully closed or anywhere in between.