How Did Europe Get Into This Mess?

Long answer here.

Short answer: in forming the European Union, it guaranteed each country a common currency and monetary policy (via the European Central Bank)... but in doing so forgot to create a common fiscal policy / manager.

What this means is that politicians in the various European countries can create more and more debt (like Greece) or can remain relatively austere (like Germany) - all individually. But if one or more of them get in-debted 'too much', the monetary authority can't do much to help them without hurting others, and since there is no one fiscal authority, these large deviations can't be prevented in the first place and long-term austerity measures can't be enforced upfront for the good of the bloc.

On a related note, unless they fix that bigger issue in the future, unlike US which has a moral hazard with helping certain industries and groups of people (auto, banks) at the expense of others, Europe actually has a similar moral hazard but pitting country against country - which is a much more scary problem.