"Feels like" is an attempt to answer a question a thermometer can't: what is this air actually doing to a human body? Two separate formulas handle it, and which one runs depends entirely on the temperature.
Above 80°F: the heat index
Your body sheds heat mainly by evaporating sweat. Evaporation only works if the surrounding air has room to accept more moisture, and humid air doesn't. So the same 90° day can be pleasant or dangerous depending on how much water is already in the air.
The heat index quantifies that. It comes from the Rothfusz regression, a long polynomial fitted to a 1970s human physiology model, and it's the standard the National Weather Service uses. Feed it temperature and relative humidity, get back an apparent temperature.
91° with 74% humidity feels like roughly 108°. Same 91° at 30% humidity feels like about 91°: no penalty at all. Seventeen degrees of difference from moisture alone.
Below 50°F: wind chill
Cold works in reverse. Your body warms a thin layer of air against your skin, and that layer insulates you. Wind strips it away as fast as you can make it, so heat loss accelerates.
The current formula dates to 2001, when the US and Canadian weather services replaced an older 1945 model with one built from actual human trials, volunteers in a wind tunnel with sensors on their faces. It needs temperature at or below 50°F and wind above about 3 mph to apply.
30° in still air feels like 30°. 30° with a 20 mph wind feels like about 17°. The thermometer never moved.
Between 51 and 79°F
Nothing to adjust. Feels-like equals the actual temperature, because neither humidity nor wind is doing enough to a body in that range to be worth modeling.
The assumption nobody mentions
Both formulas model a person in the shade. Direct sunlight can add 10 to 15 degrees to what your body experiences, and no standard feels-like number includes it. That's why a 95° heat index in the shade and a 95° heat index while standing on asphalt in July are wildly different experiences.
Neither formula accounts for your age, fitness, hydration, clothing, or acclimatization either. They're a shared reference point, not a personal readout. Useful, as long as you know what they left out.