Did they? I mean, if indeed the referendum result only answers the question of Remain or Leave, then you can't garner any more information from it than that. So the idea that you can "know" what the voters meant beyond just leaving seems a bit of a stretch.
In the meantime, we have a General Election result in between, in which support for UKIP (extreme Brexit?) collapsed, support for LibDems (supersoft vanilla Brexit lite where we don't leave really at the end anyway) didn't exactly do well either, and support for May's and Corbyn's interpretations of the question (somewhat different, although overlapping on a few issues) was split about ... well, about 52/48, funnily enough, in May's favour. I don't think there's any case to be made that the country as a whole is in favour of a "hard Brexit". Or a "soft one", if it comes to that.
In short, I don't think we have any idea what the public meant when they voted to Leave. Given the Election result, it may even be that some voters voted in favour of Leave, not because they actually wanted to Leave the EU, but to stick two fingers up to the Establishment (and Cameron in particular, perhaps).
The only way to find out would be... well, I suppose to ask the electorate again. But then maybe you'd need one referendum for each question, and you'd have to present half-a-dozen different options for each, and I'm not being serious about this proposal anyway. Politicians have to interpret the result as best they can, and I'm sure that Theresa May honestly thinks -- or honestly thought -- that her interpretation was the correct one. But ultimately it's only an interpretation, and others are equally possible. Now that she's lost her majority, it is likely that she'll have to pay more attention to other, less extreme, interpretations.