Ray Bradbury wanted readers to understand the importance of reading and thinking. One of his quotes that I think sums up much of what he was saying is, "You don't have to burn books to destroy a culture. Just get people to stop reading them." He set "Fahrenheit 451" in an unspecified, but not too distant, future. He chose that setting so that people could relate better to the idea of what might happen if people get too caught up in themselves and get too lazy to think. He speaks to the reader often through the characters of Beatty and Clarisse. Clarisse explains to Montag, in their conversations early in the story, how the society is caught up with speed and how they've become unfeeling. Beatty tells Montag in the first part, how the society came to be the way it is and this is where it is most evident that Bradbury is speaking to the reader. He warns of being too politically correct and thus watering down literature. He warns against censorship. He warns the reader about what happens when people get in a hurry and only want the short version of something and thus miss the full flavor. By having Beatty speak and showing what happens when people stop reading, Bradbury is encouraging the reader to read. At the end of the story, Bradbury gives people a ray of hope that this trend can be stopped if only people are willing to fight for the right to read and think.
Bradbury, as well as many science fiction writers, wrote his book in response to his observations of societal changes happening around him. The number of cars on the roads had increased greatly, as had the number of TVs in people's living rooms. Bradbury became increasingly concerned that we were becoming a numb society - interested in only what we could get faster and easier, interested in filling up our headswith meaningless noise. If you research the time period he was writing in, you'll be able to make many connections to what he projected for the future in "Fahrenheit 451."
Beyond societal commentary, I believe he is also making points about individuality and the person vs. society conflict.