What's the difference between "back at home" and "at home"? "back there" and "there" and ect. Why do we need "BACK"? Thanks!
"Back" implies that you've gone somewhere and now have returned.
"I've been at home all day" means you haven't left the house.
"I'm back at home now" means that you went somewhere and you are now in your house again.
"My friends went to Florida last year. We're going there this year." This implies you haven't been there yet.
"My friends went to Florida last year, and my family is going back there this year." This makes explicit that you have already been there, and are going there again.
In all of the example uses (back at home, back there, back at the offices), back is being used in a directional sense.
From Merriam-Webster's definition of the adverb back:
2 a : to, toward, or in a place from which a person or thing came • She left home and never went back. • put the book back
It's directional, because it's making a comparison between a current location and a previous one.
I am home
I am back at home.
The former simply indicates where you are presently. (A response might be, "Okay, good to know.") The latter implies that you have returned after having gone somewhere else. (A response might be, "Oh, where did you go?")
In a comment, you asked about its use in a quote from The Social Network:
You have part of my attention - you have the minimum amount. The rest of my attention is back at the offices of Facebook . . .
Here, unlike the first use, it's not an "all or nothing" situation. His attention is divided. One piece of his attention is with the speaker, but the rest of his attention is still focused on the Facebook offices.
Rather than being used in the sense of having returned, back is indicating that part of his attention is back there, where he came from.
Although the meaning is different in both cases, and it needs to be understood in context, the word back itself, is still being used in a directional sense, and refers to a location where somebody (or something) used to be. Either, as with being home, the person is there again, or, as with The Social Network, part of a person's attention has never left.
Update: Based on comments, the reason to use back is to subtly emphasize the distinction between here and there:
What were you doing back there (instead of here).
Or, with further emphasis:
It's a long walk all the way back there (and would require a second effort to return).