Firstly I must say that I am probably more biased than everybody would like on this question as I was a proponent for brexit. I know more of the advantages for leaving than I do disadvantages and I do believe this is because I research more on the leave side.
Secondly, after reading the answers above, eloquently written, fair and quite factual as they are, a point is missed although it was touched on by the first answer, indeed there is a tremendous amount of emotion involved. It's not all about economics and GDP!
In short, people care about their standard of living, who their neighbours are and their identity.
On immigration, in Britain it was much more pressing, with a population growth of 8 million since 2001, 80% of that from immigration, on a small island that offers free healthcare and subsidies to those immigrants at a net loss overall, paid for by the british population who had lived there all their lives, it's much more serious than in France but there is more to immigration than, again, money.
France is an ancient, powerful nation that has always had an extremely strong identity with their wonderful art, music, law, food, revolutions, they truly have a national identity. Just as Britain does.
Never before in their histories has Britain or France been a nation of immigrants as it is now, a popular misconception, prior to the open door policies of the EU Britain and France for their entire histories (I am taking a bit of a punt here on France, please do correct me) have never even come close to the influx we've seen in the past decade or two, which has seen immigration rise five-fold.
Remember we are talking about countries which are 1000 years old, even if revolutions have mixed things up a bit. My argument is that France's identity may be being 'watered' down as many other EU migrants flood into the country, many of whom have no desire to assimilate at all, they are economic migrants.
A country of people without a strong identity is not as prosperous as it could be, historically great nations are formed and maintained on their individual identities and the population who subscribe to it.
The other side of the argument is the EU itself. Being the colossal political union that it is, that is self electing with its presidents and committee members, that controls laws over its residing countries, many of whom have completely different cultures and identities cannot effectively rule with the same sceptre over different cultures and peoples - many people see this as bigger and more powerful government, which many people fear and have good reason to, with history at our backs.
That's another nail, as much as the EU doesn't like to see, countries within the EU do have a strong independent national identity, you only have to take the short trip from england to france, or france to germany to see how different the food, people and language is.
Economically, many professors of economics have been exclaiming for years that countries within the EU would do better as an independent, self-governing and law making state as they can reduce the tariffs that the EU set, which is around 20% on imported goods from the rest of the world, this is partly due to the mass of rules and regulations that the EU as a political union impose, increasing costs, all of these costs end up being passed down to the end consumer (me and you), the real loser is the normal working people of these individual countries, being a nation on it's own could mean no tariffs at all if the country so wishes to do meaning much cheaper goods, benefiting the normal end consumer.
As a member of the EU you must oblige to their trade rules, immigration rules and a plethora of laws that make each country effectively 20 - 30% (there is some debate about the actual figure) ruled by the european union.
The EU sounds great in theory, bringing many countries together, locked into rules that govern all of them, a single currency, matching trade rules and it was an idea for a force of good initially but the countries within this union are so utterly different it doesn't seem to work as the best solution for them individually.
I foresee Britain becoming individually more powerful now as it is free to trade with the rest of the world with its own rules and to make and live by it's own laws, as it has done for its entire history, save for the past 30 years, now not reliant on mass immigration to fill the lower end of the ladder jobs, british people will have to take that role and become more skilled and pass the money back through their own system, rather than an economic migrant who will likely send the money home to their native nation, I see the same happening with France.
My prediction is the EU will fall and soon.
With the right leaders in place and freedom to trade with the world, the good people of France and their mighty history will once again find themselves on a serious upward trajectory.
The idea that a europe of independent nation states cannot function on their own, cannot trade with eachother effectively or rule by their own law and live by in harmony is preposterous.
The world has changed since the last century and the EU can be accredited with helping this but it has far out grown its purpose.
I wonder what Churchill would say now, if he could.
I apologise for being so bias but there are facts to this as well as emotions of national identity and they are both strong.
This was my first post and I probably was too 'passionate' and opinionated, I'll try to refrain from doing that so much in future posts.