Well, lemme see what I can do. I don't know if this is significantly different from what other people are saying. I suspect not, but you tell me.
The external deficit is the difference between imports and exports plus some other stuff like net factor income from abroad.
Basically there are a set of accounts that describe all the transactions that go on in a country at a macroeconomic level called the national accounts or the national income and product accounts.
Transactions with the outside world go in the current and capital accounts. Trade and current income payments get recorded in the current account. When there is an excess of payments over receipts in this account, that's known as a current account deficit.
The upshot of a current account deficit is that the rest of the world is lending you money. (Think about it). They're not lending your government money, although they might well be doing that too. These are private transactions between private agents, for the most part--lots and lots of 'em.
To make things simpler, you can think of the current account deficit as just a trade deficit. In order to maintain a level of imports greater than your exports, the rest of the world must be willing to acquire claims against you--in other words, people must be willing to finance the deficit.
All of the European countries that are in trouble now were running current account deficits, and have found it increasingly difficult to finance these deficits. So it's the current account deficits that links them and not the alleged profligacy of their governments.
As to why would someone not want to accumulate claims against Greece, well, that's an open, big question, isn't it? Why would someone want to accumulate claims against Greece? There could be thousands of answers.
In general, pre-GFC there was a different attitude to risk across the board. Today, people no longer feel that Greece and Germany (for e.g.) represent the same or similar credit risk, and people are not willing to lend at very low rates to risky borrowers. They don't want to hold deposits at Greek banks.