Montag is a hero because he stole a book, he risked his life to read books to the women, he ran away from the only life he knew in order to save books.
Montag started out his life as an ordinary fireman, until Clarisse asked him one day if he was happy. It had never occurred to him to be happy, because people in his community didn’t really feel anything at all. Clarisse got him thinking. When the old woman he confronted in her house would rather die with her books than surrender, it got him curious, and he stole a book. It makes him question everything about his existence. Mildred, his wife, asks him why he became a fireman.
"You should have thought of that before becoming a fireman."
"Thought! " he said. "Was I given a choice? My grandfather and father were firemen. In my sleep, I ran after them." (Part I)
When Captain Beatty comes to visit him, Montag knows that this is common. Firemen lose it and steal books on a regular basis, or at least more often that he had realized. Beatty knows the signs. Their perfectly controlled society is obviously not as perfect as they thought, or firemen would not need to exist. Beatty tells him to take all the time he needs. He gives him a lecture on why society is better without books. Montag doesn’t buy it. He knows that real emotion comes from real literature.
Montag’s second act of bravery comes when he reads to the ladies from “Dover Beach.” Since he is already being watched, this is either immensely brave or immensely dumb. He is trying to tell them that it is possible to really feel.
Oh God, the way they jabber about people and their own children and themselves and the way they talk about their husbands and the way they talk about war, dammit, I stand here and I can't believe it! (Part II)
The women are upset and confused by Montag’s reaction to their inane, harmless conversation, and upset by the poetry. Of course reading it is dangerous and Montag is risking being turned in, but it is his own wife that turns him in.
When Montag is arrested, he attacks Captain Beatty and murders him with a flamethrower. He needs to go on the run from the police but he is not concerned with his own wellbeing only. He is actually more interested in taking care of the books. Still in contact with Professor Faber, he wants to be given a book to memorize, “Book of Ecclesiastes” from the Old Testament of the Bible.
"Montag." Granger took Montag's shoulder firmly. "Walk carefully. Guard your health. If anything should happen to Harris, you are the Book of Ecclesiastes. See how important you've become in the last minute!" (Part III)
All he cares about is his sacrifice not being in vain. He wants to save civilization, and save the books. That is what makes Montag a hero. He turns from an ordinary fireman, selflessly following society’s lead, to a reformer and pioneer, ready to rebuild it.