How do they determine the "feels like" temperatures for weather data?
The "feels like" value is calculated based on temperature, but altered depending on wind (cold temperatures) or humidity (hot temperatures).  Think of it this way:

1) if it's freezing cold outside, the wind will make if seem even colder because any body heat you have will be blown away quickly.
2) if it's hot outside, high humidity will make it seem even hotter because your sweat won't easily evaporate.

From wunderground:


  The Feels Like Temperatures map show what the outdoor temperature will feel like for the current day. Feels Like Index is a factored mixture of the Wind Chill Factor and the Heat Index. The combination of the heat index and the wind chill factor are denoted collectively by the single terms apparent temperature or relative outdoor temperature or simply Feels Like.
  
  Wind Chill
  
  Wind chill is the apparent temperature felt on exposed skin, which is a function of the air temperature and wind speed. The wind chill temperature (often popularly called the wind chill factor) is always lower than the air temperature, except at higher temperatures where wind chill is considered less important. In cases where the apparent temperature is higher than the air temperature, the heat index is used instead.
  
  Heat Index
  
  The heat index (HI) is an index that combines air temperature and dew point in an attempt to determine the human-perceived equivalent temperature — how hot it feels, termed the felt air temperature. The human body normally cools itself by perspiration, or sweating, which evaporates and carries heat away from the body. However, when the relative humidity is high the evaporation rate is reduced, so heat is removed from the body at a lower rate causing it to retain more heat than it would in dry air. Measurements have been taken based on subjective descriptions of how hot subjects feel for a given temperature and humidity, allowing an index to be made which corresponds a temperature and humidity combination to a higher temperature in dry air.
  
  The heat index is derived from work carried out by R. G. Steadman. Like the wind chill index, the heat index contains assumptions about the human body mass and height, clothing, and the wind speed. Significant deviations from these will result in heat index values which do not accurately reflect the perceived temperature.