Gamma radiation is a very high-energy form of electro-magnetic radiation, which occurs naturally. Gamma rays will damage or kill living organisms by alterng their DNA, so direct exposure to them is dangerous (though as they can also kill cancer cells, they can actually be used in cancer treatment under strictly controlled conditions).
The radiation from gamma rays is of such high energy that it can remove electrons from atoms and convert them to electrically charged particles (ions). The effect of this ionisation process is to kill the micro-organisms that make food decay quickly, thus slowing down the process of decay. In fact this is really a more high-energy version of the age-old method of preserving food by drying it in the sun, or the process of pasteurising milk (heating it to kill germs). Food irradiation has been judged safe by a number of bodies including the UK Food Standards Agency.

You can see a diagram of the exact irradiation process here.