No explanation, sorry. But the burning/melting was likely an instantaneous arcing phenomenon. T-stat switching may have precluded panel breaker trip. But I'll ask what brand of breaker controls the line?
Despite the apparent improbability of access, does the environ permit rodents.

My guess would be a loose connection. A loose connection would cause high resistance at that point and cause it to overheat. It wouldn't trip a breaker because there is no exessive curent draw.

Kaffine is correct.

A circuit breaker will trip if there is excess current draw - meaning an overloaded circuit or a short circuit. It doesn't trip if a high resistance connection heats up unless that causes the surrounding insulation to break down and results in a short circuit.

What you saw isn't all that uncommon on heating type appliances or anytime a circuit is cycled at near its max working ampacity. If there is a loose connection, the connection heats up and the wire expands a little. Because the wire is expanding, it also deforms slightly - as does the terminal. When the power shuts off, the connection cools and is now a little bit looser than before because all of the mating parts have deformed slightly. In addition, the heat causes the wire and terminal to oxidize. The oxide isn't a good conductor of electricity and also increases the resistance of the connection. This cycle is repeated every time the heating element turns on and off.

As the resistance of the connection increases, it becomes the "heating element" performing basically the same function as the element that's in the tank.

To prevent this type of failure, it's important to make sure that electrical connections have good surface area contact and that terminals are tight. On larger connections and aluminum cables, anti-oxidant compounds are used on the wire to prevent the oxidation cycle.

In larger installations, preventative maintenance involves scanning a bus, breaker panel, etc. with an infrared thermometer to check for "hot spots" that would indicate a deteriorating connection.

This is also why all electrical splices must be in a box. My guess is that there was nothing that would burn near that connection - and that's by design.

BTW: There are new breakers out that detect arcing - not just excessive current flow. They are more expensive, but will interrupt this type of problem if it gets bad enough to arc.

Any parts that have shown excessive heating should be replaced. In addition, trim the wire if you have some additional length so that the connection is made with wire that isn't damaged with oxidation.

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Thanks

Here is a link that might be useful: Water Heaters

Jake2007 and Kaffine...thank you for the information. In answer to Alphonse's question, the breaker panel is a Square D box...about 35 years old. The hw heater is wired with 10/2 copper on a 30 amp breaker. When I replaced the tstat and the element, I put a new lead wire from the tstat to the element terminal. The old burned lead seemed to be only 14 ga.....I used a 12 ga wire. The "burn" was confined behind the metal access panel on the tank, so there was no risk of spreading. However, it did get hot enough to melt the plastic safety covering, the wire insulation, the plastic hub on the element and the foam insulation plug behind the metal plate. I checked all the other connections and they are tight with clean copper wire behind the element screws. Thanks again for the insight into what likely happened.

And thank you for the feedback.
A good friend who is an elec. engineer said the most common and persistent cause of problems related to the scope of electricity was -loose connections. Having recently had experience with rodents chewing through wire in auto and workshop with similar results to yours,I had a momentary lapse of reason.
I asked about the panel breakers as there was a brand notorious for failure, often catastrophic. Square D has a solid reputation AFAIK.
Heater manufacturers routinely use 14ga. in this app. and do so for good reason. If you poke around the NEC it cites the basis.