They didn't. Ever.
At no point have any of their many promises - to stay in the Single Market, to eliminate red tape, to pay an extra £350m a week to the NHS, to be able to trade globally without restrictions, to continue Erasmus membership, and on and on and on - been in any danger of coming true.
So the voters voted for the most ridiculous lies, and apparently that's "democracy."
No, it isn't. It's fraud - plain and simple.
The current state of democracy in countries like UK and US brings to mind that famous Churchill quote:
"No one pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of government except all those other forms that have been tried from time to time."
The common conclusion seems to be "oh, okay, guess that's it, then" rather than "maybe it's time to try something we haven't yet".
Elections are some times a democratic institution, but not always.
Democracy is government for and by the people. Considered decision making is a very important part of it.
By fetishising voting proponents of referendums cast aside all the important bits of democracy - consultation, consideration, listening to other views, expressing dissent.... and replace it with a brutal contest of propaganda.
Voting is important, but it is not the most improtant part of a democracy and if all yu have is voting and elections you do not have democracy.
Let's start with the simple case: in a democracy, people vote to affect policy. There's a referendum, and the will of the majority of the people voting in it determines the outcome. Easy.
Now, going in a more sci-fi direction, as a thought experiment: imagine that an evil mastermind invented mind control, and used it to ensure the majority of people voted a certain way in said referendum. Technically, I guess this would still be a democracy -- but in name only. In reality, the will of a single person determines the outcome.
Pulling back into reality, obviously we don't have mind control, nor an evil mastermind concocting schemes. We do know that if trusted people in positions of power abuse that power to outright lie, and spend very large sums of money to pursue a specific goal, then the will of these people will end up having a noticeable effect on the will of the population. If someone manipulated you and convinced you of something that wasn't true for their own hidden goals, is it still really your will that you're enacting by voting for it? or have you been tricked into enacting someone else's will?
So while by a surface definition of democracy, any of the above processes are democratic because they allow people to vote, a functional democracy relies on a well-informed populace -- we knew this even in Jefferson's time ("wherever the people are well informed they can be trusted with their own government").
If you tell people R will give them S, and they vote to get S, but it turns out R doesn't give S at all and you knew it all along; that doesn't mean people voted to get R. They voted to get S, giving them R is not therefore "technically" democracy (or any other weasel way of saying it).
The current PM was one of the relatively few political voices supporting Brexit.
Seems that with enough money from oligarchs you can make enough lies stick to swing a simple majority.
We have to try and move on from this; the next decade is going to make some rich people incredibly wealthy at the expensive of many, many people becoming poorer in the UK - not just by financial measures, but in terms of opportunity, health, quality of life. It's grim.