After reading some additional articles, I think I understand it enough to answer.
The first thing to note is that the pliers holding the rods are attached to an AC arc welder (DC would work as well), which means that one side provides a positive current and the other goes to ground. When touched, it creates a closed loop for current to flow through, from the positive side to the ground which acts as a negative, also creating light in response to the passage of the electric current (the atoms become excited which releases photons - for more on that read here) - called electroluminosity.
Using just the two pliers hooked up to the current and ground, an electric arc would be created due to a slight ionization of the metals and ionization of the air in between the two nodes, but it wouldn't be as bright or as hot as using the carbon rods because steel and even air doesn't ionize as easily as carbon.
This is where the carbon comes in. Due to how easily it ionizes, ionized carbon vapor is highly luminous and also very hot - around 3,600°C, or approximately 6,500°F - which is well above the melting point of steel (1,370°C or 2,500°F) and most rock. This heat produced is what melts the materials.
So why doesn't the end result have the carbon that was turned into plasma then? Shouldn't we see it in the end?
The answer to this is that carbon evaporates easily. The carbon that is melted nearly immediately evaporates into the air and thus isn't visible as a result.
One post I found particularly useful when writing this in addition to the one Floris provided is this one on carbon arcs as well as the other articles linked throughout the answer.