The current euroskeptism is mostly due to decades of British newspapers blaming everything on the EU, and the EU having no real voice to defend itself.
Originally the political right was heavily in favour of joining the EU, campaigning for it in the 1970s. The left was more skeptical, particularly because it was seen as a potential loss of rights that might negatively affect workers. Of course, in time the EU proved to be a huge benefit to worker's right, so the left came to broadly support it.
However, in the 1980s British newspapers realised that they could easily blame pretty much anything on the EU, because the EU had no representation in the UK and rarely even noticed those articles, let along responded to them. This proved to be a big seller, offering the public and government ministers someone else to blame for all their problems and failings.
Things got worse during the 90s, when Boris Johnson (then a journalist) realized that it wasn't even necessary to distort EU stories to portray Europe in a negative light; he could simply make up complete fabrications and they would mostly go unchallenged.
For example, one of the oldest Euro Myths dates from this era. The "straight bananas" story was actually about the EU adopting British rules on banana quality standards, but was depicted as the EU imposing crazy, nonsensical and overly bureaucratic laws on the UK.
In the 2000s new countries from Eastern Europe joined the EU, and their citizens gained freedom of movement rights. The Labour government of the day opted not to adopt any limitations on migration from those countries, unlike other EU states that put temporary measures in place. The later Troy/LibDem coalition government made the same choice a decade later.
The influx of migrants allowed nationalists and racists to build up political capital. The emergency of UKIP put a semi-respectable face on xenophobia, and mainstreamed blame of foreigners to a degree not seen in Britain for decades.
And so we end up were we are today, where many members of the public know less than nothing about the EU (because what they think they know is mostly made up, ancient Euro Myths) and everything the EU says or does is perceived as malicious.