The transformer WILL be being heated. As long as the currents drawn are below the design current then the transformer will not be damaged. The transformer looks to be not more than say 400 VA in rating. He says 12V output but the current to heat the knife blade or thicker wires looks to be more than the 40 or so amps that a 400 VA 12V transformer would provide.
It may be that the transformer is more like 2V output - where a 400 VA transformer would provide 200A "safely" [tm].
And it MAY be that there is a 12V car battery hidden out of sight.
Why should or would there be?
I know not - BUT many people make many claims on the internet that are suspect or just plain untrue.
Maybe I have the Wattage wrong - if the designer pushed the iron hard and did not care about some core heating from magnetising current as the core went partially into saturation he MAY manage 1000 Watts, That's 12V at 80A+. Or 2V at say 500A :-).
The wire appears to be the winding wire - not good for 100's of amps in normal use.