When water evaporates, it has a cooling effect - this is why your body produces sweat. As the sweat evaporates, it causes your body to cool down.

The wet-bulb temperature is the lowest temperature that can be reached as a result of water evaporating.

When this wet-bulb temperature approaches human body temperature, then the human body cannot reduce its temperature via sweating. This causes your body to overheat and will eventually lead to death. This is true even in the shade, even with unlimited water to drink.

At some point in the near future, it will be so hot somewhere that the wet bulb temperature will reach 35C (95F). Once this is reached, anyone exposed to that temperature will die after a few hours. The only way to avoid it is air conditioning, and if the power fails due to the heat then that won't work either.

This is most likely to happen somewhere tropical first, but it will slowly happen in more and more places as the Earth warms.

Pretty sure they mean an event where temperature and humidity combine to the point that the human body is unable to cool itself. So your body temperature would rise and if it got high enough would kill you.

You would need some external source of cooling like cool water or air conditioning. Fans wouldn't do any good.