All objects that are in thermal equilibrium - that is, the matter and EM radiation in the objects are the same temperature - have what is known as a black body spectrum - EM radiation with properties that are a function of the temperature of that object only. That spectrum might be modified a little but atomic absorption and emission lines, but the fundamental black body spectrum will remain. The spectrum looks like the curves on this graph:
Each curve represents a black body emitter with a particular temperature, shown in Kelvin (roughly the temperature in degrees plus 273, where 0 is absolute zero). The Sun, indeed all stars have a black body spectrum. In the case of the sun, the surface, and black body temperature is about 6000K, so it looks not so dissimilar from the 5000K curve. You can see that the black body spectrum continues beyond the visible - indeed the IR part is what is responsible for heat from the sun. The earth has a black body spectrum of about 278K (5.5 Celsius), which peaks in the infra red.
So what does this have to do with the age of the universe? Well when the universe was a mere 400,000 years old, about 13.7 billion years ago, everything was very much closer together, though space was expanding rapidly, and so the universe was much hotter than it is now, so hot in fact that there were no atoms, there was just a sea or plasma of hydrogen and helium nuclei (and a bit of lithium) electrons, Electromagnetic radiation and other subatomic particles (earlier than this there weren't even nuclei, but that's earlier than we are interested in here). The universe was still too hot for the electrons to bind to the nuclei, and so photons were constantly being absorbed and re-emitted by the various charged particles that were around, and the universe was in a state of equilibrium between matter and radiation. This means there was a black body spectrum. Eventually, as the universe expanded electrons no longer had enough energy to constantly escape binding to the nuclei, and they finally bound, becoming hydrogen and helium atoms. There was still substantial interaction between matter and radiation, particularly in the form of scattering, such as Compton Scattering and Thompson Scattering. The universe continued to cool as it expanded further, and eventually cooled down to a temperature of about 4000K at which the scattering dropped off. The radiation at this point became decoupled from the matter in the universe, as the universe became transparent, though the shape of the spectrum remained imprinted on the light that passed on, and continued traveling through the universe.
In the intervening billions of years, space itself continued to stretch. Imagine drawing a wave on a balloon, and blowing up the balloon. You will see the wavelength becomes longer and longer. The same effect occurs to the radiation, but now the the very space of the universe is expanding, so photons that initially had a short wavelength, over time were stretched out so the wavelength was longer and longer, so long in fact, that the BB spectrum which peaked at 4000K now peaks at a temperature of just 2.725K - barely above absolute zero.
This temperature is the same in all directions, though there are tiny fluctuations, which resulted from small changes to the very uniform distribution of matter and energy in the early universe as we can see in the (Wilkinson Microwave Anisotropy Probe ) WMAP satellite image below. It is these tiny imperfections that seeded the collapse of the primordial matter into the stars and galaxies of the universe today.
By knowing the rate at which the universe is expanding, which we can measure from the red shift of other features of the universe such as distant stars, galaxies and quasars, and these initial temperatures (which we know from looking at hydrogen and helium in the lab) we can then deduce the age of the universe (in much the same way as we can determine how long a cup of water has been standing on a table for if we know it was boiling when it was put there)...
...13.7 billion years old.
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