HIV is a retrovirus, meaning that it has an RNA genome.
If they locate HIV DNA in her body, that suggests that the virus integrated itself into her DNA. Some viruses do that; they make DNA copies of themselves and insert them into the host chromosomes. (We actually have thousands of viruses in our DNA.)
Virus inserted into her DNA could be unable to remove itself to cause further infection. In that case, she’s most likely fine. (Unless the virus inserted itself into a gene—but that’s another long discussion.)
Finding bits of HIV RNA is more troublesome; that could mean the virus is hiding somewhere. Or it could mean that virus inserted into her DNA is still trying to make copies of itself; if the entire virus isn’t there, those bits of RNA probably won’t do much.
The HIV drugs are meant to inhibit viral replication. To my knowledge, they don’t kill virus. I can’t say anything more than that, since I don’t know what drugs they were to be able to look them up. If her viral load was low enough and the drugs administered early enough, there may not have been much virus around to damage tissue and establish a strong infection. Then, her immune system would have a better chance of controlling the virus. One reason HIV is so devastating is that it goes directly for the immune system; people can’t fight it off because the cells that would normally kill the virus have been destroyed.
Some people have immune systems that are more able to fight off different kinds of infections than other people. This little girl could have a particular immune system variation that makes her more resistant to HIV infection.
I know this is a lot of information. There is a lot going on here, which is why I would *really* like to see the original report (when they publish it).
Per petitfour's link in comment# 10:
Baby with HIV cured; UMC doctor makes history
The infant underwent remission of the HIV infection after receiving antiretroviral therapy (ART) within 30 hours of birth — Zidovudine, Lamivudine and Nevirapine. A series of tests showed progressively diminishing viral presence in the infant’s blood until it reached undetectable levels 29 days after birth. Gay switched Nevirapine with Kaletra for long term therapy.