That's a really good question, the article doesn't go into specifics.

Then the body’s own repair systems recognize the damaged DNA as foreign and get rid of it.

This is somewhat ambiguous. It could mean that human DNA polymerases see the damaged DNA, scroll backwards and forwards to the START and STOP codons, and break the bonds to snip out the bits of viral DNA. Then endogenous DNA ligases patch the ends together. It could mean that it affects DNA in the viral particles themselves (but from the context in the article I don't think this is the case). Or it could be the case that the process triggers apoptosis to eliminate the infected cells entirely; I don't think this is the case because then you have necrotic tissue all over the place, and given that we're talking about herpes viruses this means fragile skin in tender places... ouch. That's kind of like using thermite to roast a marshmallow: Fun but overkill and potentially hazardous.