How can a mirror relate Montag's world with the real world?
Your question is not very specific so it is hard to know what part of the book you are asking about. When I see the word "mirror" in connection with this book, I think about the part just a couple pages before the end where Granger talks about building a mirror factory and doing nothing for a year except making mirrors and having people look at themselves.
To me, what this means is that the society has not been encouraging people to think. It has been encouraging them to get entertainment outside their own minds -- the people should watch the walls or go to the amusement parks, but they should not be self-aware. We can see the lack of self-awareness in Millie Montag who doesn't even know that she's almost killed herself.
So the importance of the mirrors (figuratively more than literally) is that they will require people to look at themselves and think about their lives. This will, they hope, make people get to be more like regular people -- make them have emotional lives and intellectual lives again.
Maybe this is how the mirror can relate to the real world -- is that what you're asking about?