Well, part of this may be my American ignorance showing... But I never really understood what the British people got out of being part of the EU. Sure, the politicians got lots of things... But it sure looked like the only thing the average British person got from the EU was more bills to pay, and worse conditions to do it in.
In addition to what Robert wrote, environmental protection was a big deal. Back in the 70s and 80s, the United Kingdom was known as "the dirty man of Europe". I am exaggerating but imagine Flint / Michigan was a nation... Today, around 600 beaches in the UK meet clean water standards compared to less than 30 (!) that were designated as bathing waters in the mid-70s.
Mind you, the UK governments tried every trick in the book to avoid having to abide by the EU's environmental laws... But courts eventually forced them to do the right thing. Currently, the UK is the 8th largest tourist destination in the world and foreign visitors spent 22 billion USD in the country in 2015 alone. I think it is fair to assume that if UK beaches were still covered with fecal matter as they were in the 70s when raw sewage was dumped into the sea (because it was "cost efficient" and unregulated), the number of visitors would be notably lower...
Regarding the costs, the UK paid 11.6 billion USD to the EU if you subtract any rebates and money it received back. With a population of 64 million, the cost was around 180 USD annually (about 15 USD per month) and person. Statistically, the economic benefit of EU membership is estimated to be worth around 4.000 USD annually per UK household. Of course, if those economic rewards are not being fairly distributed, that is certainly a reason to be concerned. However, it would be a grave mistake to blame the EU for this as it is entirely up to the national governments to guide wealth distribution using tax as well other laws (if deemed necessary)...
Besides the effects on Britain and Wales, you also have to consider what will happen to Northern Ireland, Scotland and Gibraltar now. At least over here, the Northern Ireland conflict was a constant presence in evening news programs during the 90s. Within the European Union, it did not matter too much if Northern Ireland belonged to the UK or Ireland as long as people were able to freely move between the north and south of the Irish island. After the UK leaves the EU, this conflict might unfortunately flare up again...
Even though Robert says he is not convinced that Scotland will secede, other people are not so sure. As for Gibraltar, Spain already announced today that they would expect to take it back if the UK leaves the EU...
One other thing to consider since you are American: Until now, when American, Canadian or Australian companies decided to the enter the large EU market, the United Kingdom was a prime destination for opening European headquarters / offices due to the shared language. For many businesses from English-speaking countries, the UK was the entry door to Europe. As the UK leaves the EU, this will change and quite a few existing as well as future jobs will be moved to the continent as a result.