The economic issue here is the central role of the U.S. dollar in global economic transactions. As they write, "[O]ne of the major business lines of European banks is providing financing in dollars on
a global scale—for trade, purchasing dollar-denominated assets, or syndicating loans to corporations. Banks the world over, in fact, have a great need for dollars because much of the world’s trade, investment, and lending is conducted in U.S. currency." But during an international financial crisis, as various financial markets freeze up, it may be very expensive or even impossible at certain time for banks around the world to get the U.S. dollars they need to carry out transactions. The Federal Reserve's swap lines are a temporary measure to make U.S. dollars available at such time around the world, so that financial instability is less likely to persist and grow.
How does a swap line work? Alexeenko, Kollen, and Davidson explain:
"The swaps involve two steps. The first is literally a swap—U.S. dollars for foreign currency—between the Federal Reserve and a foreign central bank. The exchange is based on the market exchange rate at the time of the transaction. The Fed holds the foreign currency in an account at the foreign central bank, while the other central bank deposits the dollars the Fed provides in an account at the Federal Reserve Bank of New York. The two central banks agree to swap back the money at the same exchange rate, thus creating no exchange rate risk for the Federal Reserve. The currencies can be swapped back as early as the next day or as far ahead as three months.The description helps to clarify why such swap lines are not a "bailout" or any such prejudicial term. The exchange rate for the swap is locked in, and any U.S. dollar loans that are made will pay interest to the Fed. Because the U.S. dollar plays such a central role in global transactions, the Fed is just making sure that a temporary shortfall of dollars in a foreign financial system doesn't make a financial crisis worse.
The second step involves the foreign central bank lending dollars to commercial banks in its jurisdiction. The foreign central bank determines which institutions can borrow dollars and whether to accept their collateral. The foreign central bank assumes the credit risk of lending to the commercial banks, and the foreign central bank remains obligated to return the dollars to the Fed. At the conclusion of the swap, the foreign central bank pays the Fed an amount of interest on the dollars borrowed that is equal to the amount the central bank earned on its dollar loans to the commercial banks. The interest rate on the swap lines is determined by the agreement between the Fed and foreign central banks."
Here's a timeline for these swap lines. (I found it interesting that these authors differentiate between the "Global Financial Crisis (2007-2008)" and the "European Financial Crisis (2009-current)." I've been trying to sort out in my own mind, without yet reaching firm conclusions, about how to think of these episodes as connected in some ways and separate in others.)