What are hydraulically powered and electrically controlled actuators?
As wikipedia mentions, "Electro-Hydrostatic actuators" (EHA) are self contained, the only input they need is electrical power and commands.

The electricity powers up the hydraulic pump that is used to move the actuator piston, while the electrical commands are used to select what the actuator does through the use of a valve.


  
  What does electrically controlled means?
  


Means that the position (and thus the action) of the valve is controlled via a solenoid. The following scheme shows a valve with two solenoids. The valve has three positions, indicated by the three squares. Depending on the electrical control signal, the current in the solenoids changes and thus moves the selector (the block with the three squares) laterally.



Image source


  
  How does hydraulics power the actuator?
  


Through a pump. Where the fluid goes is selected by the electrically controled valve, but the piston of the actuator is moved by the hydraulic fluid.

In the image below, if the valve of the image above is in the central position, the actuator does not move.
If it is in the left position, the actuator retracts.
If it is in the right position (with the two arrows crossing) the actuator extends.



Image source