Why doesn't a high current transformer get damaged when shorted?
What you are observing here is the basis for resistance soldering. 


  Resistance soldering is the name applied to a technique where the heat
  to melt solder (or strip wire) is instantaneously generated by passing
  a high amerage electrical current through a resistive material. There
  are three key components of resistance soldering:

A specialized step-down transformer that will generate the appropriate current
A resistive material to generate the heat
The ability to complete an electrical circuit



The transformer and the leads coming out of it are copper which conducts very well. The items being placed across the copper leads are of materials that do not conduct as well as copper. That is why the various items tend to get very hot (and even melt) while the copper does not. I rather suspect that if they had placed a heavy copper bar across the leads the results might have been different.

Here is a paper Build Your Own Resistance Solderer which uses a battery charger.

Finally you may have seen one of those little "cold heat" soldering tools that run on a couple of AA batteries which came out a few years ago. They work on the same principle as described here.