And the $64,000 question is ... does graphite conduct electricity? It certainly does! The video demonstration displays this quite convincingly. Graphite is an interesting material, an allotrope of carbon (as is diamond). It displays properties of both metals, and nonmetals. However, like a metal, graphite is a very good conductor of electricity due to the mobility of the electrons in its outer valence shells.
Reviewing some basic principles of electricity we can also see why we fry the pencil. According to Ohm's Law, which is valid for most simple electric circuits, V = IR where V is the voltage applied across the circuit, R is the resistance of the circuit and I is the resulting current. Because the graphite has a low resistance and high conductivity, it is going to draw a large current through the circuit, and this large current will heat up the graphite rapidly due to frictional heating as the charges migrate through the circuit. The result is an exciting display of glowing graphite and flaming pencils.
If you've ever connected a copper wire directly to both terminals of a battery, you will have noticed how hot the wire gets. Without something with more resistance in the circuit (for example, a light bulb) to reduce the current, you're going to generate a lot of heat (and you're also going to wear out the battery before its time). This is the case with the circuit in the video. If you added a light bulb into the circuit, the current would go down, the carbon wire wouldn't get that hot, and the demo would not be so dramatic. The demonstration also indirectly illustrates the danger of a "short circuit." These exist when a resistor in a circuit is bypassed by a low-resistance wire. This can happen, for instance, if the insulation surrounding the wires on a household appliance gets frayed. If the wires meet, the current can bypass the appliance, create an extremely low-resistance branch in your household circuitry, and induce rapid "ohmic" heating and thereby the possibility of an electrical fire. It's a good thing we have circuit breakers these days!