Not a credentialed physicist, but I'll give it a shot.
The metallic magnets make contact with the positive and negative on the battery and feed electricity to the coil, that induces a magnetic field in the coil. With the magnets arranged properly, the field in the coil likely attracts the 'front' one and repels the 'rear' one, making the 'train' move, and taking the induced field with it.

(If I'm wrong, please correct me)

Oh, and *promote awesome fun science!

Promoting this video and sending it back into the queue for one more try; last queued Sunday, November 30th, 2014 4:18pm PST - promote requested by newtboy.

WITCHCRAFT!!! BURN IT WITH A RAPIDLY OXIDIZING MEDIUM!!!

Doesn't really repel/attract at the same time, but the coil does create an opposite field to the magnets, pushing it along.

Here's a better question. Will it still work if he put the battery in the other way?

Interesting.
I guess yes, in the other direction.
But will it work if you flip just one stack of magnets!?!?

Very neat idea!

If you replaced the magnets with a non-magnetic material conductively glued onto the magnet, it would still work. From wikipedia on 'electromechanical solenoid',

Electromechanical solenoids consist of an electromagnetically inductive coil, wound around a movable steel or iron slug (termed the armature). The coil is shaped such that the armature can be moved in and out of the center, altering the coil's inductance and thereby becoming an electromagnet. The armature is used to provide a mechanical force to some mechanism (such as controlling a pneumatic valve). Although typically weak over anything but very short distances, solenoids may be controlled directly by a controller circuit, and thus have very quick reaction times.
The force applied to the armature is proportional to the change in inductance of the coil with respect to the change in position of the armature, and the current flowing through the coil (see Faraday's law of induction). The force applied to the armature will always move the armature in a direction that increases the coil's inductance.

Just ask the Insane Clown Posse. Methinks they should be able to explain.

Can't wait until my boys are old enough to get off on awesome sciencey stuff like this.

I'll have to add those items to my ever-expanding "buy for the kids someday" shopping list.

I'm going to assume that this is the Lorentz force, because it's the principle that involves magnetic and electric fields. But there are setups that can use subtleties of magnetic and electric fields, it can be very complicated. Any physicist rather than astronomer might be able to explain this better... or spot subtleties.

If you notice, it only starts moving once the back magnet has touched the wire. Which i think means that the wire is used to carry the current from the battery, with the magnets providing the magnetic field for the Lorentz force to drive the train. Effectively the force is felt by the electrons travelling in the wire (F = q(E + v x B), x being vector product, cross product), but there is an equal and opposite force to be felt by the 'train'; so it travels along. If you watch, it does look like the wire is responding - i'm pretty sure the small track would have shot off to the right if he hadn't held it, and it moves as the train approaches in the longer track.

So, circuit is set up by the the wire contacting between battery terminals, current flows in a circular fashion (mostly, assuming adjacent loops don't short). Magnetic field will emanate out from the battery on average radially, i assume (this is a simplification but a reasonably safe one), so the resulting cross product - and therefore direction of the force - acts along the remaining perpendicular direction to those, ie. straight up or down the loop depending on which terminal is leading.

If you want to see how that works, you can use the right hand rule. First finger is the direction of the electron's velocity (which is traversing loops so constantly changing in a circular manner), middle finger the direction of magnetic field which always comes out radially from the middle of the coil or track, thumb F the resultant force always points along the loop - make your first finger point in all directions of a circle, keep your middle finger pointing radially out relative to your first finger, and you will notice your thumb always points the same way, no matter how v changes circularly.

It is reasonable to assume that other factors are involved, probably a current is induced into the coil as the battery moves - the battery carries a magnetic field cos of the magnets, so we then have a moving/changing magnetic field in the presence of a wire; it should induce a current which would create a magnetic field in opposition to the field of the magnets.. and so on. But i think the Lorentz force is what provides most of the motion.

All that technology and he couldn't figure out how to increase the font size on his print outs.