Why is this high amperage metal melter making an electric arc?
The assumption is wrong.

There are two parts to consider with an arc


The formation
The maintaining 


To form an arc two opposite polarity conductors are needed &amp; a sufficiently high enough voltage to initiate a current flow. This is either via ionisation of a gas between the two conductors or via electrons being liberated from the surface of the conductors

Once a current path is established via these ions an arc is formed. This arc will persist for as long as there is sufficient power behind it &amp; a current path exists.



There are a number of things that influence what distance is needed to spontainously form an arc


Pressure. Paschen's Law

Distance
Voltage 



Likewise the medium (vacuum, air, gas) and whether the conductors are pointy (concentration of electrons) or spherical.

What occurred in the video is the individual forced the current to flow via physical contact. With the current path establish an airgap was created that was wide enough that current flow was still possible.