I haven't even RTFA yet, but I was wondering if this could have applications with other viruses that become long-term residents of the body. I'm thinking of things in the herpes family like... herpes, or chickenpox / shingles. The trick with most of these is long-term, mostly-dormant viruses hiding in the cells. If you can wake them up, the immune system can clear them, but they are effectively hidden inside the cells while quiescent.
HIV is a "retrovirus", which means the the virus's DNA integrates into the host's DNA. Some other viruses do this, but I think most don't. Some are more interesting, eg EBV is a virus from the herpes family which infects several different tissue types, and we know it can integrate into human DNA inside white blood cells, but I don't think there's proof that it can integrate inside liver or stomach cells.
As a retrovirus, the HIV sequence successfully breaks into a cell, then breaks into the cell's nucleus, then into one of the nucleus' chromosomes. (This is obviously harder to detect than viruses that stay inside the cell's cytoplasm, or that enter the nucleus but stay apart as their own episome [mini-chromosome].) That's what the article is referring to when they say their method recognises a 34-base pair long sequence - it is recognising that piece of the viral sequence in our own chromosome, and then uses something to snip out enough of the viral sequence that it can no longer make new copies of itself.
Obviously you want to be careful with any therapy that involves cutting up bits of human chromosomes... :)