Gamma rays can cause atomic nuclei to temporarily deform and become radioactive. Bombarding stable atoms with neutrons can make them radioactive (generally, an atom with too many neutrons in its nucleus will emit beta-radiation). Living issue, like just about anything else, can become more radioactive if it simply absorbs radioactive atoms.
When uranium (or plutonium) is broken into smaller atoms by nuclear fission, these smaller atoms are often highly radioactive and some of them (like strontium-90) are too easily absorbed by the body, causing cancer and destroying tissue. So, when you have a fission explosion scattering all those nasty radioactive atoms like Sr-90 around, the area is going to have nuclear contamination.
The fusion reaction itself can be clean, if the fusion product is a stable, non-radioactive isotope (like helium-4). However, standard nuclear fusion requires extremely high temperatures and that is supplied by nuclear fission, which is dirty. Also, if the fusion reaction leads to the formation of unstable isotopes, then this reaction is also dirty. In fact, after the first H-bomb was tested at Eniwetok in 1952, some highly radioactive einsteinium & fermium was found in the coral reef (those elements don't exist naturally on Earth, but fusion reactions with uranium in the H-bomb created them).