The theory, apparently backed up by some experiments, is that psychopaths can make use of empathy at will, which is what allows them to employ emotional manipulation. As related in a popsci account:
For example, consider one particularly fascinating experiment by Dr. Christian Keysers, Professor of Social Neuroscience at the University of Amsterdam.
In one study, Keysers and his team analyzed the brain activity of 21 convicted violent psychopathic offenders, comparing the results with 26 men of similar age and IQ. The participants were shown movies of people hurting each other while brain activity was measured. Later, a doctoral student would slap the patients on their hands to localize brain regions connected with feeling touch and pain. The goal was to see if patients' brains activated a feeling of pain in their own brain when viewing the pain of others.
"The vicarious activation of motor, somatosensory and emotional brain regions was much lower in the patients with psychopathy than in the normal subjects," writes Keysers. "The theory seemed right: their empathy was reduced, and this could explain why they committed such terrible crimes without feeling guilt."
But one question still plagued Keysers.
How could these same individuals prove to be so charming at times?
"I remember chatting with one of the patients...a particularly severe psychopath (he had scored the full 40 points on the psychopathy checklist)," writes Keysers. "Surrounded by the guards, he seemed a most pleasant person. He was smiling, engaging, and seemed to feel exactly what we wanted from him."
So Keysers and his partner decided to let the patients watch the movies again, this time asking them to try and empathize with the victims in the movies.
"What we found was that this simple instruction sufficed to boost the empathic activation in their brain to a level that was hard to distinguish from that of the healthy controls," writes Keysers. "Suddenly, the psychopaths seemed as empathic as the next guy. Their empathy was switched on."
"Psychopathic individuals do not simply lack empathy. Instead, it seems as though for most of us, empathy is the default mode. If we see a victim, we share her pain. For the psychopathic criminals of our study, empathy seemed to be a voluntary activity. If they want to, they can empathize, and that explains how they can be so charming, and maybe so manipulative. Once they have seduced you into doing what serves their purpose, the effortful empathy would probably disappear again."
I'm not sure how well accepted this is... The BBC also ran a story on this with some additional more sketpical comments
Essi Viding from University College London, who was not involved with the study, said it was an extremely interesting finding, but that it remained unclear whether the psychopathic criminals' experience of empathy felt the same as that of the controls.
"It's dangerous to look at brain activation and say that it means they're empathising. They are able to generate a typical neural response, but that doesn't mean they have the same empathetic experience," Prof Viding told BBC News.
"We know they can generate the same response but they do that in an active and effortful way. Under free-viewing conditions they don't seem to. [...]"
There's also a newer study that seems to confirm the "empathy switch" in a larger class of "dark triad" personalities. (Also posci story on that.)