I would have thought the single main reason why American English differs so little (relatively speaking) from British English is because of the British colonization. Generally, when one country invades and conquers another, it is easier for immigrants to settle in a new territory where their mother tongue is spoken by (or imposed on) the local population. If the numbers of immigrants outnumber, and rule over the local population, they will impose their native language and customs on the newly acquired territory.
The OP could have easily asked why Australian, Canadian, Scottish, and Irish English is "almost identical" to British English.
The successful British colonization of North America began in 1607 with the Virginia company of London and lasted until 1769 with "St. John's Island"
British colonization of the Americas […] began in 1607 in Jamestown, Virginia and reached its peak when colonies had been established throughout the Americas. The English, and later the British, [the Scots and the Irish] were among the most important colonizers of the Americas, and their American empire came to rival the Spanish American colonies in military and economic might.
In 1664, England took over the Dutch colony of New Netherland (including the New Amsterdam settlement) which England renamed the Province of New York. With New Netherland, the English also came to control the former New Sweden (in what is now Delaware), which the Dutch had conquered earlier. This later became part of Pennsylvania after that was established in 1680.
The Kingdom of Great Britain acquired the French colony of Acadia in 1713 and then Canada and the Spanish colony of Florida in 1763.
Great Britain also colonised the west coast of North America, indirectly via the Hudson's Bay Company licenses west of the Rocky Mountains: the Columbia District and New Caledonia fur district
In 1980 it was estimated that over 49 million (49,598,035) Americans were of English origin, around 26.34% of the total population and the largest reported group which, even today, would make them the largest ethnic group in the United States
Eight out of the ten most common surnames in the United States are of English origin or having possible mixed British Isles heritage, the other two being of Spanish origin. Scotch-Irish Americans are descendants of Lowland Scots and Northern English settlers who colonized Ireland during the Plantation of Ulster in the 17th century.
And the majority of American men who held the most important posts and positions of power in the USA were of English origin …
The overwhelming majority of the Founding Fathers of the United States of America were of English extraction, including Benjamin Franklin, George Washington, John Adams, James Madison and Thomas Jefferson.
And as little as 225 years ago, almost half the population of the US was English. By then the English language had firmly established itself as being the most prestigious and influential language in the USA. If we look back at 18th century writings, we must admit that it has remained, fundamentally, unchanged.
The 1790 United States Census was the first census conducted in the United States. It was conducted on August 2, 1790. The ancestry of the 3,929,214 population in 1790 has been estimated by various sources by sampling last names in the very first United States official census and assigning them a country of origin. The estimate results indicate that people of English ancestry made up about 47.5% of the total population
Source: Wikipedia English American