What does the "operating range" of a thermometer mean?

The operating temperature range is the ambient (surrounding air) temperatures under which you can have the thermometer on, and it'll function properly (give correct readouts) and not excessively shorten its lifespan.

Almost all electronic devices can take higher temperatures when they're off. This is a combination of several things, including especially that the electronic components themselves generate heat, so when on they run hotter than the ambient temperature. Some are also more likely to be damaged when there is current running through them. Also, their electrical properties change depending on temperature—and at too-high or too-low temperatures, that may exceed the tolerances.

Normally you'd see a separate storage temperature quoted—but in this case, presumably they don't expect it to be stored above 122°F either. But rinsing it under 160°F water (it's IP66-rated, after all) should be fine—as long as its off. Boiling water, however, would not be. Nothing in the spec sheet says you couldn't run it through the dishwasher to clean it (again, while off)... but I'd ask ThermoWorks first before trying it...

Looks like it means that the thermometer will function between 32 and 122 F.

So, don't expect it to function if it's in your freezer and don't expect it to function properly at temps greater than 122 F.

The difference between 122 F and 190 F is that at 122 F the thermometer won't work (possibly it will work but will have reduced accuracy) but it will still be fine and continue to function properly if you drop the temp below 122. When you hit 190 F, the plastic will likely start to melt.

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).