Revealing mistake: When Mrs. Petrelli is sitting with the dead Peter, just before she lies down on his chest and starts to cry, you can see the "dead" man tilt his head slightly backwards. (00:21:10)
Continuity mistake: When Sylar comes for Isaac the painting of Hiro and the dinosaur is visible in the studio in several shots, even though Hiro took this painting to Linderman several episodes ago. The episode director even comments on it during the audio commentaries.
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 ★