This is a fantastic question actually. It'll help when I eventually need to study up on nuclear radiation effects for Volume...111. Better to get started early.
First, we need to understand why radioisotopes are dangerous to our bodies. High-energy radiation (especially beta decay and gamma decay) can damage the DNA in cells. Our skin is actually fairly good at blocking alpha and beta radiation, and a good amount of gamma radiation. But if you ingest a food containing radioisotopes (element isotopes that release radiation), the radiation isn't shielded, and the radiation is able to directly affect your internal organs and cells.
Certain radioisotopes, (example: radioactive iodine, I-131) will be collected in certain organs (example: the thyroid gland), concentrating them and increasing the severity of their effects. When radiation damages DNA, the most straightforward thing that could happen is the cell dies (this is bad). Sometimes, the DNA is damaged in such a way that cell mitosis is left uncontrolled, and that results in cancer (this is also bad). Because radiation most critically affects cell division, their effects are most pronounced in unborn fetuses or developing children.
The Chernobyl studies I sourced were most concerned with radioactive caesium (Cs-137). Radiocaesium is documented to accumulate in various hormonal glands, especially the pancreas, but also the thyroid and adrenal glands. Since radiocaesium decays through beta radiation, it will also cause abnormalities in these organs' cells. Ingesting radiocaesium is bad, basically.