How does radiative cooling between the earth and space work? On a chilly (but above freezing) temperatures on a clear night, you can freeze water outside because of radiative cooling. 
By what mechanism do warm bodies on earth actually lose heat to space? 
From the Stefan-Boltzmann Law, I know that a hot body loses heat energy at a rate proportional to the 4th power of the body's temperature minus the 4th power of the surrounding temperature. 
Obviously that isn't a complete description of what's going on though, since the temperature immediately surrounding a hot fire is quite hot as well, but you (standing far away) can still feel the radiation. 
I'm also not clear on how the temperature difference between a body and a space that is hundreds of miles away can affect the real-time radiation rate of that body? 
 A: Bodies radiate based on their own temperature and properties.  The water in your example is always radiating energy. Regardless of whether the night is clear or cloudy, it’s radiating away energy. 
But the temperature change is based on the total flow of energy: the water gets colder if the energy it radiates away is more than the energy radiated toward it.  


*

*On a cloudy night, those clouds are also radiating energy, some of which comes toward your water: those are in rough balance, and not much cooling goes on.  

*On a clear, dry night the only thing radiating back at the water is deep space, and it’s not providing g much energy; the energy radiated by the water isn’t balanced by any inflow, and the water cools. 

*On a sunny day, more radiant energy comes in than leaves, and the water can heat up.
A: Temperature is a measure of how much energy photons have $E = h\nu$ $= kT$, and the temperature reflects the energy of the photon incident on the detector. The total irradiance or EM power incident on an area $A$ is $P = \epsilon\sigma AT^4$ from Rayleigh-Jeans law.  Here $\sigma$ is the Stefan-Boltzmann constant and $\epsilon$ the emissivity of the body. For the night sky $\epsilon = 1$ and the temperature is $20K$, a bit higher than the $2.7K$ of the CMB due to stars and such. So the temperature depends on the energy of photons and their flux. Even though the night sky is far away what counts is the energy and flux of photons coming in from it. This is pretty weak and so the night sky is cold. The Earth then radiates out to the night sky a lot more than it receives from it.
