All radiation carries energy, so when radiation (here we can mean everything from light to the fall out from nuclear bombs) hits your cells it can cause chemical reactions to occur. These are not the usual chemical reactions that occur inside your cells, some of the reactions will just make the cell die, but others will make the cell start growing and dividing without any control. When this happens we call it a cancer.
Basically it’s just because radiation promotes some types of chemistry inside your cells.
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