Ultraviolet and more energetic radiation can knock electrons out of molecules. You can think of electrons as tiny magnets that enable structures of atoms to be constructed. When they’re knocked out, the structures change or break apart. Cells have mechanisms for committing suicide if they’ve sustained too much such damage, but in rare cases that mechanism, along with the one that regulates growth, breaks at the same time, and you get cancer.
Less energetic radiation cannot directly affect the electrons, but it can still generate so much heat, which is vibration in the molecules, that they change shape or shake apart entirely. This more generalized damage has less of a chance to cause the specific damage that leads to cancer, and more of a chance to kill the cell outright. There’s still some risk though. Someone recently posted in a other ELI5 how throat cancer is more prevalent in cultures that drink very hot beverages, including hot water.
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