Certain kinds of radiation can kill you by causing cancer. Radiation is actually the emittance (by unstable atoms) of tiny particles or packets of energy (aka rays) known as alpha and beta particles and gamma rays. These particles and rays are the right size to interact with the cells of your body, ripping through them, or otherwise damaging them. These damaged cells may continue to reproduce in their damaged state causing abnormal masses or tumors that disrupt normal body function.
*Radiation can kill you other ways too.. and this is a bad description of radiation.