I don't get it, in my experience I feel a greater sense of social cohesion with liberal technologists from the Netherlands, France and Germany than I do with construction workers in Northern Scotland, Lincolnshire farmers or wealthy London Soho set. I don't have a shared identity that we're all British (or English or whatever).
Possibly this is partly because I never really felt a sense of local community until I found one in Cambridge. I grew up being English in the north of Scotland, which always made me something of an outsider, and I spent most of my school-years at public school, and never really built many local social contacts.
Thus, divorced from a sense of national identity, I find it hard to see why it matters which city my legislative body sits in. I care about what it does, and that it operates in a transparent, democratic and non-corrupt manner, but not where it is. To me the supreme parliament may as well be in Brussels, or London, or Stuttgart. (FWIW, I don't think the current European administrative bodies are particularly democratic or transparent.)
Devolution of government responsibility to an efficient level makes sense, I don't believe in a heavily centralised state. At an extreme, it would seem rational to do away with existing nations and dividing Europe into more logical administrative divisions, however this is probably totally impractical because many people do retain a sense of national identity.
Why do it? The usual arguments of European peace and security, economic efficiency, and a stronger global voice for ideals that I broadly support. I freely admit that I favour this mostly because Europe follows a social democratic model, with governments willing to balance the efficiencies of resource allocation through free market economics with socially responsible regulation. I'd be far less keen on a integration with a Europe that followed either an unregulated market economy, or one with Stalinist levels of centralisation. That's not to say that I think everything is OK at the moment, but the problems are mostly about tweaking variables, not a major socio-economic change. (OK, there are a few fairly big variables where those affected will undergo major changes, mostly around things like farming subsidies and the fishing industry).