After floating on the river in Fahrenheit 451, what does Montag realize?
Montag has not had time to really think about his predicament, or the consequences of his newly-formed individuality. All his life, he considered it the job of responsible citizens to destroy books, and now he is being hunted because he stole books and read them. His escape down the river, floating on his back, gives him time at last to think about what he has done with his life, and more importantly, what he might do with it in the future:
Somewhere the saving and putting away had to begin again and someone had to do the saving and keeping, one way or another, in books, in records, in people's heads, any way at all so long as it was safe... The world was full of burning of all types and sizes. Now the guild of the asbestos-weaver must open shop very soon.
(Bradbury, Fahrenheit 451, Google Books)
This epiphany comes after he realizes that the Sun burns, and time burns as well (time continues and cannot be undone); therefore, his role as fireman was to continue the cycle of destruction. Instead, Montag realizes that it is now his job to save and remember things so the next generation has something to build from; without a base in history, people have no sense of what came before, what worked, and what failed. Montag is now going to protect knowledge and information, instead of destroying it, and he hopes to help others who might not have experience the same self-awakening.