Our Father - S3-E12
Continuity mistake: "Our Father". At the end of the previous episode, "The Eclipse, Part 2", Hiro brought Claire to the rooftop of the Deveaux building to witness Kaito handing over baby Claire to Noah Bennet. To illustrate his point, Hiro showed Claire the exact same thing happening in the "9th wonder" comic book he brought with him, and she can follow the Japanese dialogue from there. But when the exact same scene is shown at the beginning of this episode, Hiro does not show Claire the comic at all.
Chosen answer: It's not been made 100% clear. It was originally assumed that he actually consumed part of the brain, but this has turned out not to be the case. Sylar has been shown all along to have an instinctive knowledge of how things work and how to fix them - this is his actual original power and the ability that made him an exceptional watchmaker and repairer. It appears that he uses a similar process when acquiring new powers - he exposes the brain of the target, allowing him to examine it and determine how their power functions. He can then apply that knowledge to his own brain, allowing him to replicate the ability in question.
Tailkinker ★