Political necessity?

The reason why it happened is that the Conservative Party government was wildly unpopular in 2013-2014 with all of the indications being that Ed Miliband's Labour Party were going to storm the Conservatives at the 2015 General Election. Furthermore, ever since the Thatcher governments of the 1980s, the Conservatives were weakened by the 'Eurosceptic' branch of their party often being vocal, disruptive, difficult to work with, and harming the 'Not the Nasty Party' narrative Conservative Party Central HQ (CCHQ) had often tried to push in the 90s and the 00s.

Offering a referendum on the European Union therefore had two advantages:

It was a substantial, concrete policy idea that would be easy to implement and massively popular with a certain portion of the populace, not massively unpopular with the other portions, and which Labour would never offer.
By having a popular 'stamp of approval' on the European Union, CCHQ believed it would permanently weaken and weaken the difficult Eurosceptic portion of their party.

This is of course on the assumption that the referendum passed. And never let anyone tell you otherwise, David Cameron (then-PM) and George Osbourne (then-Chancellor; finance secretary and 2nd most important cabinet member) absolutely would not have proposed the referendum if they believed it had any chance of failing.

Furthermore, they assumed they'd be out of government and the referendum would never see the light of day. To the arrogant, and out-of-touch Cameron and Osbourne the policy was all upside.

As it happens, for a variety of reasons, the Conservatives actually won the 2015 General Election with a majority (whereas they were in a coalition before). And, as promised, a referendum was planned.

Ideological basis

For a substantial period of time (late 18th-century to mid-20th century), Britain was unquestionably the most powerful empire in the world. This is within living memory. The culture and expectation of Britain being a 1st rate world power is something that has only begun to fade within the past couple of generations. But a significant number of older people (people who vote) were raised and educated with the fair understanding that Britain was a superpower. For example, all of my grandparents and most of great Uncles and Aunts were being educated at a time when Britain still held all of India and most of Africa.

Since the Second World War, Britain's place in the world has unquestionably declined. We no longer have the Empire. We racked up tremendous amounts of debt to the United States. For periods in the 1970s, Britain was widely considered the 'sick man' of Europe. The feel good moments of the 1990s and Cool Britannia were quickly doused by the War in Iraq, where Tony Blair was universally seen as a puppet of the Bush administration.

Since the 1980s in-particular, life has changed for many in the United Kingdom beyond recognition. Trade unions were razed. Income disparity has skyrocketed. Town centres have become neglected. Internal tourism has been decimated. Cities like Leicester started becoming majority-minority. 2008 and the Great Recession tumbled the New Labour government and brought in a Conservative government. All parties at the 2010 general election bought into the consensus that the only way the country would survive would be to gut public sector spending. Healthcare would worsen. Education would worsen. Adult social care would worsen. Local government services would worsen.

A very large number of people came to the rational conclusion that, at least for them, their lives had gotten worse and would continue to get worse. But how does one consolidate this very clear observation with:

The Queen
Rule Britannia
Two World Wars; One World Cup
Largest empire ever known to man
The Second World War in-general, and the Battle of Britain in-particular

A lot of the media attempted to bridge this issue with a scapegoat: the European Union.

Euroscepticism

Euroscepticism first found a voice with Margaret Thatcher in the 1980s. She often disagreed with a significant number of the leaders on the continent and didn't appreciate being limited in how she could act.

Thoughout the 1990s and the 2000s, the whole media knew they could gather attention by blaming various problems on the European Union. A notable young journalist, Boris Johnson, was particularly renowned for the ludicrous and inaccurate stories he wrote on European Union directives.

The European Union was an outstanding scapegoat:

It was 'foreign'
It was 'undemocratic'
It was 'bureaucratic'

It had something for everyone. Before the result of the referendum, you'd never hear anyone defend the EU. It was seen by most of its defenders as a necessary evil in a world we could no longer rule, and isn't it nice you don't need a visa to go to Spain? No positive case was ever put forward by anyone. There was little point to. There was never any risk of us leaving.

Now, the European Union is an imperfect project. However, thanks to the economic and cultural connections brought about by the EU, Western Europe is at the lowest risk of internal armed conflict in millennia of history. Europeans are more familiar with one another than they've ever been before. Smaller states such as Ireland remain independent and sovereign but now have defenders, and allies, and representatives that allow them to assert themselves globally.

These arguments hold much less weight on an island nation, that hasn't known armed conflict within its borders since the Glorious Revolution (excluding Ireland), who within living memory had the power and the influence to dominate half the globe.

No one appreciated the EU until it was already too late. And all of the rich newspaper editors who made bank on peddling lies about this foreign government to a lost, and disaffected public thought it'd be consequence free.

Conclusion

What was it supposed to accomplish? Nothing. The referendum was never supposed to happen, and if it did, it was never meant to pass. No one with any power or influence had any idea on what to do. What Brexit would look like. What some fringe politicians had promised was an emotional return to self-government, wealth, power, influence, independence. A turning back of the clock.