Cameron may have promised (actually, more correctly, threatened - it was a failed ploy to attempt to get voters to take it seriously) to enact whatever was decided, but he had no constitutional basis to do so - no government can bind the hands of a future parliament - and he was soon gone, trotters and all, anyway.
As an advisory referendum, it wasn't subject to the same rigour that would have governed a binding one (actually, there's no such thing as a binding referendum in UK law because parliament is sovereign: see here), and courts have ruled that it would have been illegal because of funding irregularities if it had been somehow "binding". As an MP, Alex Salmond challenged this on the floor of the House before the vote, and was assured it was purely advisory.
It's gobsmacking how this glorified opinion poll has been steamrollered into forming the unquestionable Will of the People that cannot be defied, or even questioned, and how the vacuous declarations from the Leave side that there was no way that Article 50 would be triggered without a great deal of preparation and no prospect of leaving the Single Market and Customs Union have evaporated and transformed into "no deal" being the only acceptable option (itself a myth - even if we do crash out, there will have to be numerous deals, and the next decade's going to be dominated by their machinations).
And does anyone imagine for a moment that if the result had gone the other way, Farage, Cummings and their cronies would have accepted the result and not howled and acted up for the rest of this planet's career round the sun?
So the solution some want to see is presumably another non-binding referendum to follow up the original one. And so the cycle goes on and on.
If there hadn't been such cowardice in parliament and if the media had done their job, we wouldn't be in this ridiculous and horrifying situation. I've no idea how we get out of it intact.