When Neil Gaiman wrote that book burning leads to burning people, he alluded to an important theme in Fahrenheit 451: Bradbury's plea for people to have access a wide range of books to help them build empathy with other human beings and to overcome their alienation. Books expose us to other ways of living and thinking; books are, as Stephen King points out in On Writing, a form of telepathy.
In Fahrenheit 451, however, the government wants as far as possible to control people's thoughts. The fewer thoughts people have, the fewer ideas they will be able to develop to challenge the state's dominant ideology. As people's access to a broad range of ideas diminishes, so does their capacity to understand others, especially those who are different, as fully human, and fully deserving of humane treatment.
From there, especially in a society like that in Fahrenheit 451 with its strong emphasis on conformity over understanding and empathy, it is easy to move from burning the books that don't conform to burning people who also don't seem to conform to societal norms.
For a writer like Bradbury, memories of Nazi book burnings in the Third Reich, precisely to eradicate ideas that opposed National Socialism, would have been living memories, so the concept would not have seemed farfetched.
And as Gaiman points, visionary science fiction like Fahrenheit 451 is meant to be about the present, not the future. While we don't burn books in our society, Gaiman and Bradbury invite to ask in what other ways we might censor thought or fail at empathy.