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When a depositor takes money out a a Spanish bank, for example, and brings it to a German bank, in effect the Bundesbank credits the German bank, but is owed the money by the Spanish central bank, which is owed the money by the Spanish bank. Target 2 also picks up costs of financing current account deficits.
The same thing takes place within the U.S. under the Federal Reserve's payment system, except that the regional Federal Reserves clear/settle the imbalances periodically. In Europe they don't.
German claims rose 54.4 billion euros in May, the third highest on record and places its overall claims near 700 billion euros. The magnitude of the imbalance does not sit well with many German officials, but may only really be a problem if a country leaves EMU or the entire EMU disintegrates.
This exposure may be best thought of as part of the unwinding costs to Germany of a country leaving or the failure of EMU. It raises the barrier to exit, or at least raises the cost to Germany, providing additional economic incentive to keep the system afloat.