The American-English phrase neck of the woods is familiarly used to designate the place or area where someone lives.

This phrase primarily denotes a narrow stretch of wood; by extension, it came to designate a settlement in wooded or remote country.

These are the earliest occurrences that I have found of the phrase neck of the woods in the sense of the place or area where someone lives:

1-: From a letter by ‘Ocof Ocol’, published in the Richmond Indiana Palladium (Richmond, Indiana) of Saturday 28^th July 1838:

There are one great difference between us gentlemen loafers and the “Eastern gentlemen”—the “ruffled shirt gentry”—they wear ruffles on their bosoms, and we on the neither end of our shirts, as all modest men would do. I were born in these “neck of woods” and I’m not to be sceered into measures.

2-: From The Louisville Daily Journal (Louisville, Kentucky) of Saturday 28^th March 1840:

A letter from an old Kentuckian at Georgetown says—“Harrison is going it in this neck of woods.”

3-: From The Louisville Daily Journal (Louisville, Kentucky) of Friday 24^th April 1840:

To the Editors of the Louisville Journal.
Frankfort, Ky., April 20^th, 1840.
Gentlemen:—We have had a tremendous meeting at our Court House which was filled to overflowing, yard and all. Never did I hear a more forcible speech than that delivered to-day by Gen. Combs.
The loco focos tried every means to rally a meeting, but it was a failure—every body was gone to hear Combs. So the locos postponed their caucus until Circuit Court, but they will have another failure then. Harrison is our man, and many are the changes in this neck of woods for Old Tip.

4-: From The Atlas (New York City, New York) of Sunday 31^st January 1841:

We unhesitatingly pronounce the New York Atlas to be one of the best sheets that ever came to this neck of woods.—Post, Bloomington, Indiana.

5-: From the Kentucky Gazette (Lexington, Kentucky) of Saturday 17^th April 1841:

“So be it.—We learn that the “Kentucky Yeoman” has defuncted, and its edited [sic] emigrated to more genial climes. Well, this leaves but three locofoco papers in the State!! Sorry for it—want something to look at occasionally—a tough pull the three will have!”
This paragraph is taken from the Ruselville Advertiser of the 9^th inst. The Yeoman was suspended for a week or two, some three or four months ago, and the Advertiser has just heard the news. We hope our readers will not laugh at the poor fellow, for he lives in a “neck of woods” where they never get the new almanack until it is two or three years old.

6-: From the Daily Ohio State Journal (Columbus, Ohio) of Saturday 11^th June 1842:

MASS CONVENTIONS.

It is proposed to hold Mass Conventions of Whigs, during the ensuing campaign at Cleveland, Zanesville, and Dayton. Delaware has also been named—and we take the liberty of asking brother Ely, of the Chillicothe Gazette, what he would think of the expediency of one in Sugar Grove? That’s a beautiful “neck of woods,” and there are some hallowed associations connected with the spot. We’ll vouch for one thing, and that is: That the Chillicothe latch string is always in the right place.

The phrase also occurred formerly as neck of timber. For example, the following is from The Arkansas Intelligencer (Van Buren, Arkansas) of Saturday 3^rd August 1844:

For the Arkansas Intelligencer.
Bayou Menard, Cherokee Nation, July 16^th, 1844.
Friend Hatchway:—The intelligencer of the 13^th inst., has just come to hand; and in it, I see an account, or rather description of the “celebration of the 4^th” at Fort Gibson; and hoped also, to have seen in the same paper, a description of the “celebration in this neck of timber;”—the “Roman” having been particularly instructed to furnish you with the same, but he has neglected to do so, and no one else thought proper. I shall now give you a slight idea of what took place, as I have not either time or ability to do justice to a meeting so replete with social and convivial enjoyment.
First, I should inform you that the meeting was appointed and arrangements made for providing a sumptuous repast near the “old grape [illegible word]” hotel, in the shade of some fine large oaks—near one of which, gushes forth a limpid stream of the coldest water,—where, by the bye, a small supply of “juiceries” was constantly kept, to moisten the tongues of the “dry-mouthed” gentry as they come in, over-heated by a ride in the sun across the prairie. Our worthy host, the notorious “Joe Raney,” made all the necessary arrangements for the comfort of his guests […].
Now let me promise, that this was not intended as a “general affair”—but got up by a few of the “Boys” of “this neck of timber,” to amuse themselves and pass the day in an appropriate manner. […]
[…]
We next went to the spring, one and all to drink to the Orator of the day, and of “this neck of timber.”
[…] A number of Volunteer Toasts were drank, stories told, songs sung, &c. Below are a few of the toasts. […]
[…]
By Billy D-y—The characteristic of the “neck,” swift horses, and pretty girls; few in number but “hard to beat.”
[…]
By Phila. Bill.—Toby Small our absent friend—Unlike the rest of “Old Rips” children, may he be welcomed to the bosom of his nativity, in a manner becoming one of his standing in this “neck of timber.”