A Euro Basis Swap is simply a derivative product that allows the holder in the case of a Euro swap the ability to swap EUR for US dollars.
In simplest terms the more negative the value the greater the demand for USD. As witnessed by the rapid decent the central banks were forced to act when they did to facilitate those no longer able to access USD through normal market operations.
The central bank swap lines allow banks to now work directly with their central banks who in turn swap currencies with each other. Below is a chart that shows