When you start the first coil swinging on and off the end of the first magnet, a current is induced in the coil. Since the two coils are part of the same continuous circuit, this current also flows through the second coil.
A current-carrying coil of wire behaves like a magnet. The magnetic field around the second magnet attracts or repels the second coil, setting the second coil in motion. (Alternatively, you could say that the magnet exerts a force on the current flowing through this second coil. However, the electromagnet explanation is simpler.)
When the second coil swings, it becomes a generator too—that is, a current is induced in the coil. The resulting current in the two connected loops is a result of both coils swinging through both magnetic fields.
Reversing the coil or the magnet's polarity changes the direction of current induced in the coil. This in turn changes the direction in which the other coil swings.
The clip lead short-circuits the coils. The electric current generated by the coil you are swinging will not flow through the second coil, so the second coil will not move.