I think the key take-away of Brexit is the now undeniable rise of populism in Europe. The strategy of European elites, both on the left and on the right, has been for the last decades to insult or simply ignore the segments of the population that hold "backwards" views. This is not going to work if the segment reaches a certain threshold.
The Brexit vote was a rejection of elites in general. The economic elites, from industrialists to bankers, were heavily in favor of staying in the EU. The cultural and political elites as well. And they lost.
More left-wing people will paint this as a reaction against "neoliberalism" and inequality, for which the EU is (wrongly) blamed) More right-wing people will see this as a reaction against the snobbish and holier-than-thou attitude of the leftist elites, who accuse everyone who thinks different of being a bigot. But I think there are multiple dissatisfactions, not only in the UK but also in France and other European countries:
-Large segments of workers are indeed unhappy about the pace of globalization and liberalization, which they see as resulting in less job security, stagnant or falling wages, and worse working conditions.
-These same sectors are completely alienated by mainstream left-wing parties, whose main concerns now are simply to push for mass immigration, multi-culturalism, environmentalism and LGBT issues. If anything, this segment of the population tends to be hostile to illimited immigration, which they see as further compressing their wages, and multi-culturalism, which further attacks their "old way of life".
-Simultaneously, nobody is very impressed by the far-left, which is offering in 2016 solutions which were tried and failed in 1917, and is just as likely (if not more) as the mainstream left to go on crusades about multi-culturalism, LGBT issues and other stuff that is very far from the heart of the working poor. Which is why the far-left is disappearing in countries such as France.
-Hence, votes like Brexit, which is essentially a huge shot in the foot, but also a middle finger to elites of all political persuasions. Also, the rise of the far right in continental Europe.
-La solution? Je ne sais pas.