When we bring a magnet towards a coil, a current is induced. As the magnetic field is generated around the coil and there is interaction of the magnetic field lines with the external field, energy is stored in the field (similar to electric field lines). As soon as we stop moving the magnet, the field goes away. Where does the stored energy go?
When the magnet is brought closer to the the coil , magnetic field due to the magnet in the region of the coil changes with time and this gives rise to an electric field or an electric field is induced in that region . This induced electric field applies force on the electrons inside the coil and they start moving which gives rise to electric current and thus they create their own magnetic field.
Now let's see this from the perspective of energy : the moving magnet was the source of energy in the region of the coil and this energy was stored in the form of induced electric field . Those electric field lines transferred their energy to the electrons for moving them around the coil (since the induced electric field as shown in fig does a net work on them since the angle between the force and displacement is zero here. )
The energy which electrons get is mainly released in the form of heat due to the resistance of the coil and very little amount of the energy is in the form of induced magnetic field. So when the magnet stops moving the energy supply to that region stops and all of the energy gets used up by the electrons of which major part is released to the environment due to resistance and since the induced magnetic field disappears and energy can't be destroyed then this means that all of the energy is dissipated in the environment .