When the train is placed completely within the copper coil, the circuit is closed. A current flows from the +ve end of the battery, round the local part of the coil, back into the -ve end of the battery (the magnets are conducting).

This circulating current generates a magnetic field. Proper orientation of the head and tail magnets leads to both magnets being pushed in the same direction because of their interaction with the generated magnetic field.

This is the force that drives the train.

Magnets are serving two purposes; First one as contact point for the coil and completing the solenoid, and the second one to push the battery. Metal coil is generating magnetic field.

http://en.wikipedia.org/wiki/Solenoid

http://en.wikipedia.org/wiki/Ampère%27s_circuital_law

I'm not sure but it looks like the same effect of a homopolar motor: http://en.wikipedia.org/wiki/Homopolar_motor