The popular tale has it that Newton had the idea while lying in the garden at his home in Woolsthorpe between the years of $1665$ and $1667$ and watching an apple fall from a tree.
This tale has the feel of an off-the-cuff comment that Newton may have made during the course of an interview with a journalist of the time, or whatever the equivalent may have been.
The children's version of the tale has it that the apple hit him on the head.
However, the inverse square law was in fact conjectured by Edmund Halley in $1684$, but he was unable to do anything to prove his conjecture. He discussed this with Christopher Wren and Robert Hooke, who claimed he had a proof of it. However, Halley disbelieved him.
Some months later, Halley had the chance to ask Newton what law of attraction would cause the planets to move in an elliptical orbit. Newton answered immediately that it would be an inverse square law, and claimed to have already calculated it.
Some sources state that the question was posed the other way round: that Halley asked how the planets would behave under a central force obeying the inverse square law.
The mathematical work that Newton he had performed to prove had supposedly been deduced from Kepler's Laws of Planetary Motion.
Having been thus spurred on by Halley, Newton went ahead to write and publish his Philosophiae Naturalis Principia Mathematica.
