This question has recently become of great interest to me, having moved from the lovely, temperate Bay Area to a part of the world where the weather is actively trying to murder me for nine months a year (fun Minnesota fact: the hot/muggy summers are actually worse than the Road-esque winters in many ways!) Even so, I never really put much thought into what the "feels like" temperature field in ForecastFox signified, besides an excuse to bitch about the weather even more than I already do.
There are two major methods of estimating what temperature it "feels like" outside, based on the actual temperature, relative humidity, wind speed, and various other things. When it's hot, your go-to metric is the Heat Index; in the cold, it's the Wind Chill Factor. I'll cover heat index today, and probably get around to wind chill tomorrow or sometime later this week.
The heat index was originally known by the inexplicably-awesome term "humiture," since it mostly bases its calculation on temperature and humidity. However, since there's no way to say the word "humiture" without sounding like the language center in your brain just broke, the National Weather Service started calling it the Heat Index when they adopted it as a standard around 1980 (Canadians, who always kind of sound like their language center is broken anyway, persist in calling it the "humidex" to this day).
The basic heat-index calculation is both conceptually and mathematically simple. The basic deal is that your body has a built-in, somewhat gross air-conditioning system known as "sweating", where it basically shoves a bunch of salt water out of your pores and lets it evaporate, cooling your skin via evaporative cooling. Since evaporation rate is inversely proportional to the amount of ambient water vapor already in the atmosphere, this otherwise pretty clever setup isn't going to work as well on a humid day. So as a result of your body's reduced ability to cool itself, your perceived body temperature is going to be quite a bit higher on a humid day vs. a dry one, all other things equal. I've illustrated this in my chosen medium of MSPaint below.
The heat index, before it became a way for bored weather forecasters to scare people, was designed to take this into account for not-dropping-dead reasons. Basically, the heat index is the air temperature scaled by the ratio between the partial pressure of water vapor in the air (humidity/dewpoint, essentially) and an arbitrarily-selected baseline humidity of 1.6 kPa, or about 0.01 atmosphere. So in essence, if the humidity is above that threshold value, the heat index will be higher than the actual air temperature; below it and it ends up lower.
Meteorology is where linearity goes to die though, so there are a whole bunch of caveats and exceptions to this seemingly-simple relationship depending on what the heat/humidity conditions are like. For all intents and purposes though, the heat index relationship is considered to be useful at temperatures above 80F and relative humidities above 40%. The effect of wind is ignored entirely, so a rigorous (ha ha) "feels like" temperature measurement will take both heat index and wind chill into account (which you can't do because they're valid in totally different temperature ranges -- see tomorrow's post for details).
There are a whole bunch of assumptions baked into the heat index relationship, mostly about human-body stuff like mass, height, and blood thickness, as well as some sketchier assumptions like "amount of clothes worn" and "level of physical activity." Since anyone who has met more than one other human being knows that there is quite a bit of variation in all of these things among the population, you've probably gathered by now that the whole "calculation" isn't a whole lot better than a random guess unless you're a statistically-average adult human from 1980. Wikipedia, in its inimitably understated style, says that "significant deviations from these [average parameters] will result in heat index values which do not accurately reflect the perceived temperature." Still, it was created with the best of intentions (the aforementioned keeping-people-from-dropping-dead-of-heatstroke) and can still be more useful than the raw temperature when you're trying to decide whether or not today is a good day to run a random marathon.
FUN READER EXERCISE: The heat index was developed shortly before America became a nation of disgustingly overweight slug people. Try recalculating the heat index using the difference between median adult weights in 1980 vs. now and find out what it "feels like" to ride a Rascal down the street to Burger King on a hot summer day! Or just get fat and try it yourself, I guess, if you don't like math.