This is one of those fluffy topics which is easy to dismiss because it is difficult to show any concrete proof that it has worked or is working. Absence of war by itself can always be argued to be coincidental.
But consider that the EU has its origins in a plan to make former enemies so interdependent that they would never consider going to war against each other again.
Therefore the EU is at base, a peace plan. Leavers often say that all we need is a trade agreement, nothing more (well perhaps a few common standards too). But the history of the EU explains why it is much more than just a trade agreement. The fact that it is a peace plan is what makes the EU unique and not comparable to other trade pacts such as NAFTA and ASEAN. Don’t forget that the EU was awarded the Nobel Peace Prize in 2012, for its contribution to “over six decades… of the advancement of peace and reconciliation, democracy and human rights in Europe.” Yes, it may have been politically motivated but there must be some truth in the claim.
What about NATO?
Some people argue that it is NATO that has kept the peace. But NATO is a defensive organisation armed only with weapons.In fact, in some areas of the world NATO is not seen as defensive at all but as the aggressor. Long before you get to the stage of armies (or their technological proxies) facing each other off, there has been a human breakdown in communication. And this is part of what the EU tries to do; to promote interaction, contact, communication, collaboration, cooperation and any other C-word that you could care to come up with!
Our insurance for peace
Peace-building is a process that requires active construction which is what the EU tries to promote by financing international projects in research and education at all levels. Peace-building needs to happen at a person to person level as well as at an inter-organisational level and international diplomatic level. As Churchill said:
to jaw jaw is better than to war war
For me, the peace-building trumps financial cost to a very large degree. Think of it as insurance. I also think that the EU needs to do more in this area especially in the area of increased migration (including but not restricted to refugees and asylum seekers). The more we can get to know each other and involve each other in our ordinary lives, the less conflict there will be.
No, nobody can prove that the lack of wars between EU countries is proof that the EU works. But if you want to vote leave then I challenge you to explain how you will ensure these fluffy contacts will be maintained, encouraged and increased because they are vital.