There are a number of different types of magnetic security tags. There is very little literature describing them, probably because the manufacturers don't want everyone to know how they work, since they then might be able to deactivate them themselves. However, one article on the subject, written by R. C. O'Handley of MIT, appeared in the April 1993 issue of Journal of Materials Engineering and Performance, page 211.
Usually there is a magnetic dipole antenna pair at the exit to the library or store, which exposes the tag (and the customer) to an alternating high-frequency electromagnetic field. It is of course important that the system produces a response only from the security tags and not from any other magnetic materials the customer might have, such as paper clips or hairpins. There are several different systems to produce a special magnetic response from the tags. Sometimes the sensor looks at specific harmonics (higher frequencies) of the magnetic response, which will be different for the tags. Others depend on resonant effects; the size and shape of the tag is designed to match the frequency of the antennas and create a much larger response than other magnetic materials. Many magnetic materials will respond very differently to electromagnetic fields when magnetized to saturation than when demagnetized, which might give you a clue as to the deactivation process. But if you learn too much more, you might be able to deactivate the tags yourself!
The free shiny metal strip is a magnetostrictive material made of amorphous metal ("metallic glass") which exhibits a relatively high amount of magnetostriction (10-6 or so). Basically this material expands and contracts in response to a magnetic field. Just like a tuning fork or any other strip of stiff material, this strip has a mechanical resonant frequency. When you tune the AC magnetic field to this frequency, you get a big response. This signal is then detected by the pickup coils, either as a resonance or as a ringing after a pulse of magnetic field is applied. The other piece of metal (rhomboid) is a permanent magnet. It turns out that the magnetostriction requires a small DC field to activate the coupling between magnetic and mechanical. So when this strip is demagnetized, the resonant strip no longer "sings" and the tag becomes dead.