Power Transfer Switch Burning Up

Mandalay Parr wrote:
The connection was probably not screwed down tight.

x2; this is the most likely cause, in my estimation. Fix it (and replace the overheated piece of wire at the same time). If the transfer switch itself shows signs of overheating, like discoloration or melted/blackended plastic bits, it would be wise to replace it or at least those components if they're individually replaceable...or eliminate it.

Some sort of a transfer switch is needed whenever you want to feed the same circuit from two different power sources (at different times)--in this case, from the shore power cord and from either a generator you don't have or perhaps from an inverter. If it's wired to a generator you don't have, and you do not foresee installing a generator, it would be perfectly reasonable to eliminate the transfer switch entirely.

Overloading your 20A supply, while a bad practice, would not have caused this particular failure.