They believe that the people who run "Brussels" are a group of corrupt elite that want to take over their countries, in a authoritarian dictatorship. Or they simply do not like centralization, they believe that they will lose control over the issues that concern them and won't have direct say in the decisions that will affect their lives.
Though the issue of centralization is a valid one, many European nations are actually a federation and they deal with this issue effectively. It is actually nation states that lack behind like Greece, when all power and money is gathered in the capital. In the UK, Germany and Switzerland for example, they gave more power to the regional capitals and that is the way forward.
In a globalized world, no nation or government is actually independent or sovereign. All countries must follow the developments or suggestions of the Markets if they want to achieve prosperity. No country can exploit their natural resources without allowing the investors that follow the Markets' ratings to invest in the state! So how independent are we really?
I will remind you that Europe after two World Wars was totally destroyed, its economy was in tatters and it needed cash to kick-start production again. That money come from the Marshal Plan that was set up by USA! Millions of dollars were poured into the European economy and while it helped the Continent's recovery, it also made Europe attached to America until now. But as many of the plan's critics have said, it promoted corruption in the Governments that received lump sums of cash.
Did you think that Europe would receive that money for nothing? The Marshal Plan helped to transform Europe and its policies in almost everything, but it started the "Americanization" of the continent and most of the nations that got involved. Colonization ended because of American intervention and European economies became more open and globalized because it was a requirement in order for a country to be part of the plan.
That is why the Americans were keen to give as much money to as many countries in Europe, even to the eastern block and Asia. South Korea, Japan, Pakistan were also receivers of American money. No wonder they are to this day close allies of USA. The eastern European block rejected the Plan so they remained controlled by the Soviets.