Generally, this means a district, neighbourhood, or region. In conversation, it usually means “around here” or “in this vicinity.”
Since the 14th century, people have used “neck” to describe a variety of things that were narrow or constricted, like the top of a bottle, a mountain pass, an inlet of water, the fingerboard of a stringed instrument, and so on. So the early colonists were merely carrying on a tradition when they used “neck” to describe a narrow piece of land.
The usage was first recorded in colonial property deeds. The Oxford English Dictionary’s earliest citation is from a document written in Dedham, Mass. in 1637: “Graunted to Samuell Morse yt necke of medowe lying next unto ye medowes graunted unto Edward Alleyn.”
In Early America there was apparently a conscious attempt to depart from the style of place names used in England for thousands of years. So in place of moor, heath, dell, fen and similar Old World terms, the colonists came up with branch, fork, hollow, gap, flat and so on. Though “neck” had been used in Britain for things like mountain passes or inlets of water, the Americans were the first to apply “neck” to a narrow stand of woods or, more importantly, to a settlement located in a particular part of the woods. In a country then largely covered by forests, “neck of the woods” was the first American neighborhood.
The original word “neck” dates back to the 800s (it was first recorded in Old English as hneccan), and comes from old Germanic sources.
“My neck” might also come from the German phrase “meine ecke,” which means “my corner.” No one can verify such transformations, but they are possible. For example, this one: “Don’t be fresh” transformed from German “frech” which means insolent.
That would certainly make a lot of sense to me. As a teenager, I was often told not to be “fresh” which puzzled me, because the literal English meaning of “fresh” seemed to have no connection to the bratty comments I made.