When the magnet approaches, N flux inside the bifilar coil increases. An opposing flux is produced by the coil and an EMF is generated across L1 and L2. The EMF across L1 applies + voltage to B-E terminals of Q1 and turns it ON. Q1 saturates rapidly and a high current is withdrawn from B1. This current builds a strong magnetic field inside the coil. Since L1 and L2 share the same core, an opposing flux is produced by L1 and the magnet is pushed away.
As the magnet goes away, the flux inside the coil decreases. An EMF is induced across L1 that applies - pulse at B-E terminals and Q1 is turned OFF.
Consequently, collector current is cut sharply. As the magnetic field inside L2 collapses, a high voltage spike is produced by L2 that charges B2.