A dying fire does not “cool” you...it just doesn’t warm you as much. The heat radiates out (well, convects and conducts by a campfire, but radiates with the earth) and is not replaced as much...but the sun doesn’t “cool”... That’s why the phrasing seemed odd.
We are getting caught up in semantics. You're right, the sun doesn't cool you directly, for that to happen it would have to be colder then we are, but it does cool as it radiates less energy. Its surface temperature goes down as its activity decreases and therefore it is cooler then it was during a more active phase.
The heat radiates out (well, convects and conducts by a campfire, but radiates with the earth)
You had it right with "radiates", unless you are sitting directly above the fire there is very little convection transferring heat to you and the same can be said for conduction. That leaves radiation as the primary mode of heat transfer. As the fire dies it also radiates less heat so that you do indeed cool.
Think of it this way:
You are pumping water with an old fashioned hand pump into a bucket with a hole in the bottom. The water level in the bucket depends on how big the hole is and how fast you pump. The effort you put into pumping represents the sun's energy input to earth (the bucket), the hole represents the energy lost from earth by radiating into space. The water level represents the earth's average temperature. If you think about it most any level of pumping will result in a stable water level, the faster you pump the deeper the water gets. That is as long as the sun lasts the earth will continue to be warmer then the cosmic background (although not necessarily in the range that supports human life!). It's activity level determines our average temperature, just as pumping harder will raise the water level. Radiative heat transfer depends on the absolute temperature difference between heat source and heat sink. The flow of heat is just like water, it always runs downhill. The only exception is when we input energy and pump it back uphill (refrigerators, air conditioners, and yes, heat pumps). The sun (at about 11,000° Fahrenheit radiates to us at room temp and in turn we radiate to the rest of the universe at −452°F.
Some think the universe will eventually end when all matter has reached the same temperature and no further thermal energy transfers are possible.
Regards,
GtG