Why do so many bitcoin exchanges have their bank accounts closed and have problems establishing bank accounts?
Countries have AML (anti-money laundering) laws that limit activities that could enable money laundering. A Bitcoin exchange arguably can be used for money laundering. Even if the letter of the law doesn't say anything specifically relevant to Bitcoin, banks don't want to take chances.

To be able to respect AML and other laws, banks do need to know a few things about the owner and the business, and they have a "know your client" procedure for it. For most businesses this goes uneventfully, for others (e.g. Bitcoin exchange) it could be a problem.