I'd rather be independent as a nation (whether that be Scotland or the UK) than be a US-like state within a larger, wider nation. Obviously Brexit wasn't about actual independence (since we have that at the moment), but I don't think it'd be a particularly good idea for the majority of Europe to be beholden to a government which, for many of the countries, could be a continent away with a massively different culture/style of government.
Well as someone who'd like to see some kind of federal Europe, the thinking is that as everything from companies to terrorist groups start to operate on an international scale, governments should do the same to ensure that the people in control are the governments which have some control on them from the voting public. While Europe would (and should) never be as centralised as the USA due to the vastly different culture and circumstances, I think there should be some powers handled at the Europe-wide level because I believe it would be more effective.
Besides larger countries like China and the USA are all going to look after their own interests and have the capacity to pressure the relatively smaller European countries in doing things they might not want to. Only by banding together can they stand up to those outside pressures.
Regarding the democratic accountability thing that some have brought up, that actually annoys me. The UK has a monarch as head of state, a completely unelected upper house which includes 26 bishops and 92 peers who inherit their seat (inheritance is often limited to men only apparently) as well as a prime minister that is chosen by a couple hundred thousand party members at best and a few hundred MPs at worst and a cabinet that is chosen by that prime minister. Not to mention how many of them come from a handful of schools seemingly.
The British political system wouldn't have looked out of place in times before the EU even existed. British people don't have much more influence on British politics than European people do on European politics.