The high speed fan is commanded when AC is requested, coolant temperature exceeds a certain setpoint, or when the PCM (powertrain control module) sees no signal from the sensor. This happens to ensure that to ensure the car doesn't overheat. The engine temperature is used to determine where the gauge points and when to activate the appropriate cooling fan speed. When those two symptoms are seen together (the malfunction indicator light, or MIL - also known a the 'check engine light', will usually be illuminated as well), it is a result of the expected operating temperature is not achieved after a specified period of time. Low coolant level will cause no coolant to be present at a GOOD sensor (so the expected temperature sensor will indicate a temperature significantly lower that the that of the engine), and a failure in the coolant temperature sensing circuit will cause the signal to the PCM from the sensor to be completely absent (as a result of a BAD sensor or the wiring between it and the PCM). From what I can see, there's no relay - everything needed is part of the cooling fan (the image below came from resource on my public library's website).
View attachment 70255