The mechanism is theorized by some to be that the radiation stimulates the immune response, exercising and strengthening the mechanisms which manage the ordinary tissue damage that occurs every moment of every day due to oxidants, other natural metabolic waste products, and natural cell death.
That is a claim, not an explanation of how the effect works.
Ionizing radiation (sufficient energy to break covalent bonds) leads to odd chemicals inside the cells. The energy of ionizing radiation is absorbed in breaking the covalent bonds. If the bond is in DNA, then you can get a cancer, or you can get a mutation, which may be good, or may be bad, or may make no difference. If it is not in DNA, but in a tissue, you have miniscule tissue damage that is repaired. If it is in a chemical that is not tissue, you have an odd chemical in a dose that is too small to have any effect.