As described in this article at Wikipedia, in 1666, Newton was in a garden when he noticed an apple falling. Surely, Newton could not have been the first person to notice that things fell when dropped. I'm sure many gardeners saw that happen and I found evidence from old stories about Galileo Galilei dropping items from towers and other stories about people struggling to build winged machines which seem proof enough that people noticed that things tend to go down. So why was Newton's observation of this behavior so much more significant?
The observation that apples fall to the ground is not significant in itself. What matters is the conceptual jump that Newton performed, while (as he reports) sitting in his garden. Before Newton, the conundrum was expressed as: "if apples fall, why does the Moon stay in the sky ?". The breakthrough was Newton suddenly realizing that it was the wrong question: the Moon is also falling toward the Earth ! But it has enough lateral movement to "keep missing" (that which we now call "being in orbit").
Newton could think that because he had already come up with the realization that when an object moves, it has, by itself, no reason not to keep moving: you have to push a trolley to set it in motion, but you also have to do something to stop it. Prior to Galileo, everybody was convinced that a moving object could not keep moving indefinitely; every object, by itself, had to stop. Galileo was first to demonstrate that it was not true -- actually, the first to come up with the idea that trying things out was a better way to demonstrate things in physics than just thinking about them abstractly. Newton, with new mathematical tools ("infinitesimal calculus" -- an idea that Leibniz also got independently at around the same time), could formalize it as his first law. With that knowledge, he could imagine, finally, that if the Moon had some lateral movement, then it would keep on having lateral movement and thus keep on missing the Earth while it falls towards it.
It all crystallized in his head at some point where he was, mostly, at rest in a quiet environment. As Nietzsche is believed to have once said: "all truly great thoughts are conceived by walking." It is thus kind of fitting that Newton made one of the biggest advances of physics of all time while observing something as mundane as an apple falling. Such is the way the human brain works.
(Incidentally, the apple story has been used to pinpoint the exact date of the scientific breakthrough, since apples fall only at a specific period in autumn. However, the story is a nice story that was reported only by Newton, who knew how to sell his ideas, so the story might be a myth after all.)
To add to @Thomas-Pornin's answer, and Ben Crowell's comment, Newton understood that a force was acting on the apple. The question was how high the apple would have to be before the force stopped acting. As in, why doesn't gravity act on the moon? Newton's insight was that the force never stops acting, it is a universal force.
Newton understood gravity long before the apple fell before him from the tree -- his explanations, however, came across as the mindless drivel of a lunatic to the common man of the time.
Surely for the unfortunate genius Newton, coming up with a way to get these people to understand something so obvious to himself was the hard part!
The apple falling from the tree was the story that finally struck a chord with his audience for long enough for them find value in trying to understand why Newton was trying to tell them.