In Fahrenheir 451, when Montag explains to Faber that "my wife's dying," what does he mean?

I assume that you are talking about the part where Montag
first goes (on the subway) to talk to Faber.  He wants Faber to be his teacher.  Faber
says that Montag is brave and Montag replies by saying that he's not brave but that his
wife is dying, a friend has died, and another person who could have been a friend died
as well.  When Montag says that Millie is dying, he does not mean it literally. 
Instead, he is speaking about Millie's soul, her
humanity.

Montag is trying to explain why he stole the
book.  He is trying to tell Faber that he did so because he was driven to it by how
desolate their society is.  That's why he says Millie is dying.  She is dying inside
because their society does not want people to think or to have normal human feelings for
one another.  The ban on books is a part of this.

So Montag
is explaining why he stole the book.  He is saying that their society either kills
people literally (Clarisse, the old woman) or kills their humanity (what has happened to
Millie).  Because their society does this, he feels he must fight back and that is why
he has stolen the book and come to Faber.