Explain the following quote: “So it was the hand that started it all . . . his hands had been infected, and soon it would be his arms . . . his hands were ravenous."

When examining a quotation from a story it's always best to look at the context in which it was used. In this particular case, Montag has just returned home from another book-burning expedition. This one was especially unpleasant as it involved an old lady burning herself to death in her home as an act of defiance against the regime. Before he left the scene, Montag helped himself to one of the few books that wasn't consumed by the flames. This is what he means when he says "So it was the hand that started it all;" it was his hand that picked up the book, as if it had a mind of its own. And what the hand started was Montag's rebellion against the state. Reading that book will be the first step in Montag's subversive journey from loyal state functionary to implacable foe of this totalitarian regime.
When he gets home, Montag feels the exciting poison of disobedience running through his whole body. It's as if he's eaten forbidden fruit, as Eve did when she ate of the tree of knowledge in the Garden of Eden. His hands still seem to have a mind—and appetite—of all of their own. They are "ravenous," that is to say very hungry; hungry for knowledge, truth, ideas, all the things that are considered dangerous in this conformist society. And it's not just Montag's hands that are hungry for the book and all that it contains; his eyes are equally ravenous, burning with passion to "look at something, anything, everything."