I think they’re referring to what is commonly called the ECTS or engine coolant temperature sensor. EFI means “electronic fuel injection”. The tech is correct about what the ECTS does; the engine computer uses the output of that sensor – along with other sensors – to determine how much fuel to inject in order to get the correct air to fuel ratio. And if it is bad, that would indeed confuse the computer and could cause the engine to run rich, possibly enough to stall out when warm.
A couple of things to consider though. First, if the ECTS has completely failed, that would usually cause a check engine light. Second, if it hasn’t failed but had become inaccurate, that would usually also turn on the check engine light b/c the computer would observe (via the O2 sensor) that the engine was running unexpectedly (in your case) rich or lean. I’m a little doubtful about this diagnosis – unless there are other clues – for example stored diagnostic codes in the computer memory consistent with the above.
If I suspected a problem with the ECTS on my Corolla, the first thing I’d do, I’d remove the part and test it in a pan of hot water with an ohm meter. I’d do a two point test, one at ambient temperature, and one at 180 degrees F. On my car anyway that part is a thermistor that screws into the cooling jacket, a device that changes resistance as a function of temperature. The calibration chart to compare to is provided in the shop manual. The ECM measures the resistance and by that knows the coolant temperature.
Best of luck.