A hybrid car uses power from both internal combustion engine and high voltage batteries for propulsion.
Transmission fluid is a special fluid which is used to lubricate and also to cool a transmission.
There are different types of automatic transmissions and different specifications of automatic transmission fluids, depending on the car. Most hybrid cars use eCVT (electronic Continous Variable Transmission) which uses planetary gear sets and electrical motor/generator sets.
The electrical machine in the transmission acts as a motor when it is being used to drive the wheels and acts as a generator when it is being used to charge the battery.
Since the motor/generator units are electrical devices, they use special automatic transmission fluid that is designed for use in hybrids.
Reason why hybrid car transmission require special fluid
Toyota hybrid synergy drives for example require WS (the short form of world standard) transmission fluid. This is because it is specially formulated to have low electrical conductivity. Normal transmission fluids will conduct electricity and cause various forms of malfunction of the hybrid system such as erratic charging, vehicle going off by itself while on motion and damage to the transmission itself.
When a wrong transmission fluid remains in the eCVT, it begins to show signs of weak high voltage battery and eventually, the battery fails. In severe case, the hybrid car begin to shut off on motion. Although there are more than one reason than can cause a hybrid car to be shutting off on motion.
The best maintenance practice is to always use the transmission fluid that is specified by the vehicle manufacturer.