In essence, an antibody called N6 was extracted from an HIV patient. The patient is a long-term nonprogressor who has had HIV for 21 years, but an exceptionally low viral load of <1000/mL and normal CD4 T-cell count. This indicates that his immune system has kept the virus at bay for all these years with essentially no ill effect.
HIV viruses have a surface protein complex called Env, which binds to the CD4 surface protein to gain entry to T-cells (part of the immune system). HIV cannot transmit without this protein complex, and so certain parts of the complex are very well conserved across different HIV strains.
N6 binds to this part of Env, in some sense "simulating" CD4. However, because it's just an antibody and not a cell, once Env is bound it cannot infect anything. This is what makes N6 a neutralizing antibody.
The thing that makes N6 special (more potent and more broadly applicable) is that it is able to avoid the parts of Env that HIV doesn't need for replication. Imagine Env as a wrench. The head of the wrench is a specific hex shape to mate with CD4 and can't change. Antibodies can grab on to the wrench handle to prevent infection. But the handle can change, and HIV regularly mutates this part to avoid being caught - adding side branches, changing the shape, and so forth. N6 is cleverly arranged so that it can strongly bind to the wrench without grabbing the handle too strongly - in other words, it tightly binds the head of the wrench like a person wrapping a thumb and index finger around it (this hand analogy appears in the paper). The result is that HIV has a much harder time getting out from under the antibody, since mutating itself to alter the head shape would also affect infectivity.
Now that larger HIV screening trials are done, and the technology has advanced to better detect such super-antibodies, more of these clever antibodies may be discovered. We may yet have a good shot at defeating this tricky virus!