There are several factors that go into this. They produce sun screens for themselves to protect against the sun, they produce antioxidants to try to mitigate DNA damage, and they have powerful DNA repair mechanisms compared to animals.

All of this pales, however, in the face of their anatomical resilience. Plant cells are immobile and no part of their body is irreplaceable, except the primary stem and even that is only irreplaceable in some species. The parts of a plant that are most exposed are typically leaves or photosynthetic stems, and in most cases these are only retained for a few months to a few years before they fall off. In the event that plants do have uncontrolled cell division, it cannot metastasize and they will probably be rid of it soon. If it's on the main trunk, it still probably isn't stopping anything essential.

So they do have more powerful mechanisms for DNA repair, but this is enabled because if they have a catastrophic error it's not actually catastrophic while in animals a catastrophic repair mistake means death. Animal cells will often die if their DNA is notably damaged rather than risk becoming cancerous while plant cells are less likely to do that and so they benefit from being good at fixing things.