This colloquialism is used for describing people's opposition to an official plan. Although they support the idea, they do not want it to happen in their neighborhood or, anyway, in the are where they live

Everybody agrees that waste disposal is an issue that must absolutely be solved, but when the township planned to build a waste incinerator the whole community's reaction was: "Not in my back yard, you don't!"

Tutti dicono che lo smaltimento dei rifiuti è un problema che va assolutamente risolto, ma quando la municipalità propone di costruire un inceneritore il commento unanime è: "Non a casa mia! Vicino a casa mia non ce lo voglio!"

The phrase seems to have appeared first in the mid-1970s. It was used in the context of the last major effort by electric utilities to construct nuclear-powered generating stations, especially those located in Seabrook, New Hampshire, and Midland, Michigan (Encyclopedia Britannica).