Incoming links: Kelvin
This is not a truly "fundamental" constant of nature like say the speed of light or the Planck constant.
Rather, it is just a definition of our Kelvin temperature scale, linking average microscopic energy to our macroscopic temperature scale.
The way to think about that link is, at 1 Kelvin, each particle has average energy:per degree of freedom.
For an ideal monatomic gas, say helium, there are 3 degrees of freedom. so each helium atom has average energy:
If we have 2 atoms at 1 K, they will have average energy , and so on.
Another conclusion is that this defines temperature as being proportional to the total energy. E.g. if we had 1 helium atom at 2 K then we would have about energy, 3 K and so on.
This energy is of course just an average: some particles have more, and others less, following the Maxwell-Boltzmann distribution.
If you point a light detector to any empty area of the sky, you will still get some light.
The existence of this is quite mind blowing, since "there is nothing there emitting that light".
To make sense of how it is possible to see this light, you can think of the universe as the expanding raisin bread model, but it expands faster than light (thus the existence of the cosmological event horizon), so we are still receiving light form the middle, not the borders.
CMB is basically perfectly black-body radiation at 2.725 48 K, but it has small variations with variations of the order of 200 microKelvin: cosmic microwave background anisotropy.
If you adda bit of impurities to certain materials, at low temperatures of a few Kelvin their resistivity actually starts increasing if you go below a certain critical temperature.
Kondo effect graph for gold with added impurities
. Source. 
For scales from absolute 0 like Kelvin, is proportional to the total kinetic energy of the material.
The Boltzmann constant tells us how much energy that is, i.e. gives the slope.