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When Airfoil Satellite for Mac is being used with audio transmitted from iOS or iTunes, IPv6 networking can prevent a reliable connection.
This problem seems to be specific to a network configuration where all devices on the network use Automatic IPv6 networking, and the receiving Mac is running Mac OS X 10.9 (Mavericks) or later. iOS devices and iTunes attempt to connect to Airfoil Satellite for Mac, and fail with a warning that they cannot connect.
Setting link-local or manual for the entire network, or just the individual sending/receiving devices (using Network settings) can allow Airfoil Satellite to reliably receive these connections going forward.