The government, for its own reasons (internal splits, fear of UKIP taking votes), decided to treat it as binding. (Side note: had it been a legally binding referendum, it would have been declared illegal because of irregularities in how the campaign was conducted.)
No planning was carried out before the referendum to figure out what the impact of Brexit would be, how to cope with it and how to enact it. Civil servants in the various departments were actually forbidden by Cameron from committing any thoughts or plans to paper.
The Leave campaign was deliberately vague and contradictory about what form Brexit would take. Some of its prominent supporters insisted that it wouldn't mean leaving the single market, customs union etc., just a less formal trade-only relationship with the EU. Some treated it as a big bluff, expecting the EU to fold and give them whatever they wanted because the UK was indispensable to the EU - they insisted the UK could have its cake and eat it, basically being able to trade with the EU without having to comply with its rules.
May, on taking over from Cameron, interpreted the result as primarily meaning that freedom of movement (in the EU sense) would end, having been obsessed with immigration and her consistent failure to bring numbers of immigrants down during her term as Home Secretary. That red line was mainly what led to all the problems in the subsequent negotiations. The EU will not budge on the "four freedoms" principle (of free movement of goods, services, capital and people) that governs its closer trade relationships with countries that want access to the single market. Other than that, May continued to be vague about the shape of Brexit, coming out with platitudes such as "Brexit means Brexit" and that she wanted a "red, white and blue Brexit".
To make matters worse, Article 50 was triggered prematurely, before the UK had carried out any planning, even to the extent of what its aims in negotiations with the EU would be. Now, at this more than late stage, it's no clearer, because the splits in the country, parliament and the two main parties are so severe that no feasible deal can carry a majority, and any time efforts are made to nail down a path forward, it leads to stalemate. Last year, May finally called her Cabinet to Chequers (the prime minister's country retreat) to try to thrash out an agreement on what shape Brexit should take. Within barely a day, this triggered resignations from a number of her prominent ministers who had second thoughts. The resignations have continued since then as the mess has unfolded further.
The main sticking point - and a totally predictable one from the outset - was the issue of arrangements for the Northern Ireland-Ireland border (the interface between UK and EU jurisdiction, and an obvious potential flashpoint). That still hasn't been resolved, no deal is possible until it is, and even a no-deal Brexit wouldn't resolve it.
Basically, the prospectus adopted by the hard-line Leavers is undeliverable. Now they're reduced to spouting about the virtues of a no-deal Brexit and the wonders of trading on WTO rules, which would be severely disadvantageous to the UK economy, and the problems of which none of them seem to grasp, or if they do, they're lying about them.
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