What is a listening squawk?
A listening squawk is a transponder code you set to indicate to an ATC controller that you are listening out on a particular frequency. 

Unlike squawk codes assigned by the controller, the listening squawk is not unique to your aircraft but is a set (and published) code for the particular area. In the UK it is commonly used in the uncontrolled airspace around a major airport's terminal control area.

Examples in the UK include the airspace surrounding Luton, Stansted, Leeds, and one dedicated to the narrow "low level corridor" allowing VFR flight between the class D airspaces of Manchester and Liverpool.

The procedure is: after signing off from whatever facility you were previously in contact with, set your com radio to the assigned listening frequency and set your transponder to the published listening squawk. Do not contact the controller, instead listen out for any broadcast which may be intended for you (eg the controller may ask the aircraft in a particular location to respond). They will know you are on frequency because of the code you are squawking. 

The controller will generally only attempt to contact you if you are about to violate controlled airspace or have a traffic conflict. Note that this should be treated as a backup only, the responsibilities for navigation and see-and-avoid remain as ever with the pilot.

Once out of the area, or when you need to change frequency to your destination or next enroute contact, turn the transponder back to the VFR conspicuity code 7000.

If I recall correctly my instructor told me that this is a UK/CAA-specific initiative and is not found in other countries.