It started in the 16th/17th centuries. In the 1650s "you" was being used for "social superiors" despite it being a plural word. Quakers resisted that change, instead maintaining use of "thou" for singular (even to social superiors) and "you" only for plural. At that point, this was highly unusual. It was one of many snubs they gave to the ruling class, and they spent a lot of time in jail.

In the 19th century, Quakers were still using "thee" but by the American Civil War rarely "thou" (odd grammatical quirk particular to Quakers, "he is", "she is", "thee is"). Everybody else was using "you" regardless of listener's status or place in the sentence. By this point, "thou" was so unpopular in the English language, that you could tell if someone was a Quaker by whether they used thee/thou or you.

I am, however, hazy on when precisely (between 1650 and 1850) the switch from thou to you for your social inferiors came to be popular. I'm pretty sure it went region by region, because "tha" (thou) is still used in Yorkshire, England (and several other nearby regions).