What mechanism is used to measure ocean currents from cables?
The most likely explanation is that a device similar to a hot wire anemometer has been used; but one that has been designed to be used in oceans. 

A hot wire anemometer is basically an electric heater that has been calibrated. The heating element radiates heat into a surrounding medium, depending on the design and  construction of the anemometer, either air or water. The greater the heat loss, the greater the flow of the cooling medium over the anemometer element.

By calibrating the anemometer it is possible to determine the velocity of the medium, in this case ocean current, over the anemometer and from that it is possible to determine the volume flow rate.

Solid state devices, using silicon transistor sensors have also been developed to measure ocean currents.