Several times throughout the book, Montag recalls meeting Faber in the park, although it is not made clear why this is important. He is reminded of Faber by several things, including his first meeting with Clarisse and most significantly, when he is sick and Chief Beatty is visiting. Montag feels the air blowing from the wall vent and thinks of meeting Faber; it turns out that he has been stealing books for a long time and hiding them in the vent. When the meeting is finally described, it is indicative of the police state in which Montag lives (and willingly participates); Faber is terrified, but subservient, and only after a long time does Faber open up and recite a poem from memory.
Faber held his hand over his left coat-pocket and spoke these words gently, and Montag knew if he reached out, he might pull a book of poetry from the man's coat. But he did not reach out. His. hands stayed on his knees, numbed and useless. "I don't talk things, sir," said Faber. "I talk the meaning of things. I sit here and know I'm alive."
(Bradbury, Fahrenheit 451, Google Books)
This is part of Montag's move towards individualism; he realizes that a person like Faber is useful in society but that Faber has been marginalized and his purpose destroyed to promote the government's agenda. Montag contrasts his memory of Faber's conversation -- one of the only meaningful conversations that he has ever had -- with the superficiality of Mildred's TV programs, and resolves for the first time to take real action.
Montag meets Faber in the park. After a short discussion revolving around small talk subjects (like the weather), Montag learns that Faber was a former English professor, who was thrown out into the world after the last liberal arts college was closed.
When Faber becomes more comfortable with Montag, he begins to speak excerpts of poetry, with his hand placed carefully over his pocket. Montag believes that Professor Faber most likely has a book in that pocket, but does not press the issue.
The two talk for an hour, Faber speaking mostly in monologues, poems and pieces of literature. Montag never tells the Professor that he is a fireman.
Faber leaves Montag his contact information, which Montag later uses when he discovers his love for books.