According to Hugh Rawson's book "Wicked Words" the sexual sense of "laid" is an Americanism. Usage in print dates back to at least the 1930s, so usage in spoken word was likely earlier.
It's almost certain that the use of the term derived directly from the past tense of "lie," that word having been used for hundreds of years as a way to describe sexual intercourse. In the Bible, for instance, Lot's eldest daughter said, "let us make our father drink wine, and we will lie with him, that we may preserve the seed of our father," and then she went in and lay with her father. Shakespeare made similar use of the term.
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