As far as I know, radiation destroys enzymes in our body, so these changes could harm us. It depends on the amount of exposure; a bgger exposure is worse. Or radiation could make us into Spiderman (who was bitten by a radioactive spider) – but not really!
Radiation causes damage to the cells of our body, and especially to our DNA. While there is some DNA repair machinery in our cells, it cannot cope with too much damage, so the cell will either die or there wil be errors introduced in the repair process. These DNA mutations can have all kinds of negative effects, including increasing the risk of cancer and (if the mutations affect DNA in our germ cells) the risk of having children who are disabled or ill.
It’s dangerous because some radiation is very high energy, and sometimes in the form of ionised particles, and can cause damage to cells. But for the same reason radiation is useful in cancer therapies. We are mainly afraid of it because it something we can’t see, and can’t necessarily stop..