In addition to the articulate answer of the first poster, this particular quote reveals Mildred's distress at this moment in the text. The repetition is a result of her inability to form a complete thought. Remember, Mildred's entire life is lived in front of those television walls. The fact that she refers to the actors as "the family" shows that she considers them a bigger part of her life than she does her own husband. Watching the house burn is like watching her family die, & the repetition becomes her only way to express the sadness and grief she feels. The fact that "everything" is repeated as well shows that she is essentially stuck on this thought: she cannot comprehend that her life as she knew it is over.
So repetition is used both for emphasis and to express the character's distress. It almost sounds like a skipping record (or mp3, I suppose now), like Mildred's mind is incapable of processing the information entering it. She can only articulate the same few words over & over again. This also reveals what was truly important to her. As she watches her house burn, she doesn't think about the items she & Montag may have purchased together, or perhaps their wedding album. Instead, she focuses on pieces of glass, metal, and fiber optic cables that are easily replaceable. So, this repetition also exposes her materialism.
I think that the most obvious answer to your question is that the writer is using the literary trick of repetition. Repetition is the equivalent of hammering a nail into the wall and then giving it one more whack for good measure. It is a way that the author can call emphasis to something the he or she wants to make certain that you understand.
Notice, in the quote you give, how the first "poor family" is repeated twice; in addition, "everything gone, everything, everything gone now" repeats "everything" three times and "gone" twice. Ray Bradbury, in writing this, wants to be certain that you are made away of how the character is feeling.
Be careful when using this trick, though. It is like a spice that can be easily overdone. Repetition can "hammer things home," but when used too much it can knock the sense out of the reader. In addition, don't confuse this creative use of repetition with repeating yourself in other types of work. In general, repeating what one is saying in an essay is a big no-no. It makes you look as if you have run out of things to say. In short, if you are going to repeat a phrase or set of words (as Bradbury does here) make sure you are doing it for a good reason.
On a side note, notice the way that Bradbury, in the quote you mention, is able to convey the impression that the speaker is in a difficult frame of mind. His repetitions, while providing emphasis to what he is saying, is also indicating that he is "shell shocked." He sounds like a man muttering incoherently and in a rambling fashion. That's good dialogue!
Anyway, good luck with your writing analysis!