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There is radiation which can be split among alpha (helium nuclei), beta (electrons) and gamma. The first two are physical particles and the last is a high energy photon. They are dangerous when they hit you but have no residual danger. Once an alpha or beta particle has hit something and lost all of it kinetic energy it is just a free helium nucleus or electron.

The danger is from radioactive materials which produce these types of radiation. For example iodine-131 will decay into xenon-131 which is chemically inert and a beta particle. If you suck down a big glass of water with a lot of I-131, it will concentrate in your thyroid, emit a lot of beta radiation and potentially kill or give cancer to your thyroid. If you boiled that glass of water all the I-131 would stay there just as salt would when you boil salt water, so you would get even more I-131 when you drink enough water to quench your thirst. If you had a full distillation equipment you could get the iodine-free steam off the boiling water and then cool it down to make your glass of safe water.

I was taught that radioactivity has no physical form but rather attaches itself to stuff. Dust, for example. By boiling the water you remove the dust and any other particulate matter that is capable of "holding" the radioactivity.

Have you ever boiled a pan dry? Or even just let drops of hard water evaporate? The foreign matter (unless it is in a form more volatile than water, like alcohol) will just concentrate in the pan as you boil the water. Where I live that means a nice crunchy layer of lime (yum, yum).