"He acted from loneliness. That's usually the reason behind the choice. I was the first in Carlisle's family, though he found Esme soon after. She fell from a cliff. They brought her straight to the hospital morgue, though, somehow, her heart was still beating." –Twilight, 252
What Edward didn't know, primarily because Carlisle himself didn't realize it until much later, was that it was Esme who had put that loneliness there in the first place.
The teenager who'd fallen from the apple tree, stubbornly refusing to cry more than absolutely necessary, who'd confided that she'd broken her parents' rule about climbing to help her kitten down from the tree, and had been happy that the cat's landing had been much more graceful than her own, had really been the beginning of it. Her warmth was enough to reveal to him what he'd been missing.
Of course, Carlisle had been lonely before, especially after leaving the Volturi, but the loving girl, with her broken leg and content kitten dozing in her lap had brought on a fresh wave of loneliness. A wave that would persist longer than any other had, making his decision to leave harder, despite the fact that he was already pretending to be thirty five. It was that same wave that still haunted him seven years later, when Elizabeth Masen begged him to save her son, and made it so much easier to do as she asked.
So it was thanks to Esme that they had Edward. And Edward, who was more their son than any of the others in that strange way, didn't even know.