I agree with what /u/dawnquixotee wrote, that this is probably a better question for /r/linguistics. To answer briefly in that context, this is mostly a myth. As linguist William Labov says in part of his conclusion in The Social Stratification of English in New York City:

All of these results put together support the remarkable finding that great cities of a million or more are geographic unities. Any given neighborhood may differ linguistically from another but such differences are correlated with socioeconomic class differences in the population. Though to say so loses all credence in the public forum, it must be asserted that “Brooklynese” is nothing but a convenient geographic label for working-class New York City speech.

In other words, there aren't different accents specific to New York City boroughs, but there are different accents specific to different social and economic groups of people--black New Yorkers, white New Yorkers, Italian-American New Yorkers, Hispanic-American New Yorkers, Jewish New Yorkers, upper class New Yorkers, lower class New Yorkers, middle class New Yorkers, college-educated New Yorkers, less-than-high-school educated New Yorkers, New Yorkers born before WWII, New Yorkers born after 1990, and so on.

The whole book is dedicated to the subject, and Labov studied many different groups of people from many different New York City neighborhoods, and came to the conclusion that the only correlation to geography is that particular neighborhoods tend to be populated by particular socioeconomic groups. But there are many different socioeconomic groups in every borough, so there aren't really borough accents, or neighborhood accents, either. An upper class black New Yorker born and raised in Brooklyn sounds like an upper class black New Yorker born and raised on Staten Island, and a lower class Jewish New Yorker born and raised in the Bronx sounds like a lower class Jewish New Yorker born and raised in Manhattan. It might seem there are more geographic accents in New York City, but that's just a result of there being so many people of so many different socioeconomic classes living in a relatively small area, and there being socioeconomic differences between boroughs--there's more upper class New Yorkers living in Manhattan than in Queens, for instance. In any case, it's not really a geographic phenomenon.

So there's not really much to say on the subject from an historical standpoint. It's more of a linguistics question--why different socioeconomic groups living in a related geographical area speak in different ways. You'll probably get a more thorough and informed answer at /r/linguistics.