PSA bounces aren't well understood. There is a research paper which mathematically models the effect of the immune system mopping up old cancer cells which fits the observation. That isn't proof this is the cause, but is the best guess we have.
What we do know is that people who have PSA bounces tend to have better outcomes, and it tends to happen in younger patients with better immune systems. PSA bounces are usually hidden by hormone therapy, and so only seen on those not on hormone therapy after radiotherapy.
It helps to understand that radiotherapy doesn't immediately kill all the cancer cells, but does render them unable to divide and multiply, and hence no longer malignant. Those that survive the radiotherapy will then go on to die of old age or be mopped up by the immune system over the next couple of years. This isn't normally how cells die - they normally kill themselves (apotosis) when just past their best so your body doesn't have a lot of old cells in it, but this function is usually broken in cancer cells, leaving them to die much later of old age.