Recently, bitcoin fell because South Korean government threatened to ban trading. What I fail to understand is how can cryptocurrency trading be banned? Citizens can always trade anonymously in overseas exchange. What can governments do about it if they cannot track the anonymous trading that takes place?
They can't entirely ban trading for all of their citizens worldwide, but a government can ban trading within the country itself. This means that it will be much harder for people to trade and thus drastically reduce the number of people within the country that are trading.
However while you can trade anonymously, any earnings made from trading are usually taxed. Not paying your taxes (even for illegally earned money) is a good way to get a government coming down onto you.
Furthermore, in order for someone to even use the money that they earned, it must go through the banking system which governments usually have control over. Thus they can force banks to make it difficult, if not impossible, for people to deposit and withdraw their money from exchanges.
Without any way for people to get their money out of an exchange, most people won't change since trading anonymously just means that any earnings you get are almost entirely unusable to you.