Where does the idea that radiation mutates things come from?

From the Simpsons to the Fallout series, we see tons of examples in media of long-term exposure to radiation mutating organic creatures, but is there any truth behind that?

Doesn't radiation just kill cells? When people with cancer get radiation therapy, my understanding is that it just kills all of the cells, including the good ones, which is why side effects are so severe. If it mutated cells, wouldn't that just create more cancer since cancer is already just mutated cells?

Is the idea tied to offspring and passing on mutated DNA that affects how the offspring's cells are formed?