Microbiologist here (so no cancer expert!)

Some molecules are more stable than others. As you already mentioned, DNA can be damaged by radiation as energy is taken up by the DNA molecule and then it can fall apart or create new bindings.

Long story short, food molecules can (during their metabolism) turn into highly reactive particles that can damage your cells and DNA. Especially oxygen radicals can cause a lot of damage.

There's gotta be a YT video explaining this a bit more in depth.