For example, they need to know where do you get your money from, and if it's from mining, they want copies of bills for the mining rigs, and so on.
But you have to have robust systems first.
There is also the huge problem that they do not have MSB licenses in the US, so it's only a matter of time until Florida or NY sues them, or demands the arrest and extradition of the owners for not complying with US laws yet still allowing American customers.
Foreign services stopped dealing with US residents because last year new FBAR and FATCA compliance rules went into effect, requiring US taxpayers to provide more information about their foreign assets, and the US signed numerous new agreements with most major nations to share data about U.S. account-holders (agreements under which either nation could demand specified information about account-holders in the other nation as if they were domestic institutions) Many European banks stopped doing business with Americans because it was a paperwork nightmare to deal with the compliance.
The Final Rule requires each foreign-located MSB to appoint a person residing in the United States as an agent for service of legal process with respect to compliance with the BSA and its implementing regulations.
Translation: Bitstamp, if they take $1 from an American customer are now required to register with FinCen, possibly apply for licenses (nobody has figured this out yet at the bitcoin foundation) and have an agent based in the US to oversee legal compliance. I haven't heard of Bitstamp doing this. Use at your own risk.
Just like you don't worry about crazy laws in Saudi Arabia or Kuwait that prohibit drinking alcohol.
They can try and press the UK, but that, at best, will only result in Bitstamp moving its headquarters to a different country, which would result in a ton of taxes leaving too.
They can and do go after banks that have subsidiaries in the US. They also can pressure those banks that don't by going after their affiliates that do.