It’s because it mostly damages the DNA in the cells, but leaves the rest of it more or less alone. You don’t notice until the cells try to divide and can’t because the blueprint to make a new cell has been destroyed.
It’s the same reason sun exposure causes skin cancer. Sometimes it fucks the DNA up just enough that it forgets when and why it’s supposed to divide or let itself die, so it just starts making a bunch of extra cells with little rhyme or reason.
EDIT: to add in because someone else mentioned it, the immune response with the inflammation and heat happens because now the body has a bunch of dead and dying skin cells it needs to clear out and replace. It might even have some precancerous stuff it needs to try and catch and destroy before it becomes a much bigger problem.