This question has been asked and answered already. Along with the answer here, see those at the link in the Sources section.

A major theme in this novel is alienation. Mankind has alienated itself by destroying books, books which contain the record of the history of mankind. When men are alienated, they are lonely and depressed and they wind up finding no reason to live, so they commit suicide, like Mildred. It is significant that Montag decides to memorize something from the Bible, because the Bible is the word of God, given to man so that man will know how to live in the world. The particular book that Montag tries to memorize is Ecclesiastes. The reason that this is significant is that this book is written by a wise philosopher, some believe King Solomon himself. In this book, the author grapples with the meaning of life, man's place in the world, etc. The author experiences everything life has to offer (fame, fortune, family) and finds them all "meaningless, chasing after the wind." The conclusion of the book of Ecclesiastes is found in Chapter 12, verse 13:

The conclusion, when all has been heard, is: fear God and keep His commandments, because this applies to every person.

In Fahrenheit 451, man has strayed far, far away from his purpose, and thus he is totally alienated. But, there is hope - God's word will not return void and can be stored up in a man's heart through memorization, which Montag decides to do to redeem himself.

Ray Bradbury is one of my favorite authors of all time. While he often cringes when people point out his religious themes and motifs, one cannot deny their existence in almost all of his works. He was raised a Baptist so these themes are a part of the fiber of his writing, despite his denials.

Read about it here on eNotes.

I might also add that it is ironic for him to memorize scripture because this action is so emphasized in the modern church and across all religious beliefs.  I doubt there is a single world religion that has not encouraged the preservation of their holy texts through the art of "hiding it" in one's heart.  If the Bible is the single most widely published book in the world, I wonder if there is evidence to prove that scripture is the single most widely memorized text.  Food for thought.

The Bible is a primary source for the ethical values of Western civilization.  The Bible is the source of basic concepts such as love of one's fellow man, refraining from stealing, murder, and incest, belief in one supreme Being, belief in an after-life, and prayer.

It is therefore not surprising that Montag, the hero of Fahrenheit 451, chooses to memorize parts of the Bible in order to preserve it from the flames of the "firemen."  By memorizing the Bible, Montag would be preserving many of the core values of civilization that were quickly being forgotten in the empty, carefree, and careless society in which he lives.

The specific book that Montag chooses to memorize is the book of Ecclesiastes.  This is a difficult book that contains many pessimistic thoughts about the vanity and emptiness of life; Ecclesiastes urges us to focus on wisdom and simple pleasures, rather than foolish extravagance.  This book would be a good antidote to the headlong rush to hedonistic emptiness that was taking place in Montag's society.