Basically they couple the antenna to the one side of the mains cable, normally by either wrapping an insulated wire around it a few turns or by using a 10pF 1kV ceramic disc capacitor. The metre or so of mains cable before the wall outlet then is the antenna, with the board forming the other half of the monopole antenna, or really a very poor half of a dipole. Can work well for FM in high signal strength areas, but has poor performance that is affected by other equipment in the house plugged into the same outlet.