Humidity as others have said, but another big reason is radiative heat transfer.
Heat transfers to and from us in a few ways. Convection through the air is typically the most significant – air moves over and around your skin, and if it’s cooler than your skin it will remove heat, cooling you.
Radiation is another way heat is transferred. Radiative heat transfer is always happening. To keep it ELI5, all surfaces constantly radiate heat to some degree. Typically a hotter surface will transfer heat to a cooler surface. When you’re outside on a 60 degree day, the most significant source of radiative heat is the sun, but even in the shade this can be significant. The heat from the sun reflects off surfaces and heats you, and as the sun heats objects they begin to radiate heat to you as well.
Inside, if your AC is set to 60, the walls are probably also 60, so the net radiative heat is leaving you, cooling your skin.
The sum of these effects is the temperature it “feels.”
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