What does the "operating range" of a thermometer mean?
It's the range at which it will display an accurate reading.

Or in some cases, any reading.

Update : I guess the terminology is different for this thermometer, as it says "Guaranteed accuracy ±2°F to 248°F" suggesting that it can handle 248°F ... so I'm guessing that in this case, it's talking about the ambient air temperature.  (and it's possible that the 190°F limit is like the storage temperature on computers -- you can let it get to 180°F, provided you let it get back to the allowed operating range before use).