This is a fairly "common" issue, that typically occurs due to high resistance at the terminal. The high resistance can be caused by corrosion, or a loose connection. The high resistance causes heating, which causes higher resistance, which leads to more heating. Eventually the insulation on the wire starts to melt, and/or burn. Usually the heat will transfer through the terminal block, and into the smaller wiring in the stove. When the heat becomes too much, the smaller appliance wiring will fail and open the circuit.
In most cases this is not a super dangerous event, as the wiring is contained within the stove. However, there is a chance that this can lead to a fire in the house, especially if the cover is not properly installed.
Causes
Corrosion
Corrosion is more commonly a problem if the wiring is aluminium, though can still be a problem with copper wiring. Because stoves/cooktops/ranges draw large amounts of current, there is typically some heating of the wiring during operation. The constant heating and cooling cycles of the wiring can lead to corrosion, and/or hardening of the wiring over time. This corrosion and/or hardening increases the resistance of the wiring, which cause the heating to be worse.
Solution
Occasional inspection of the terminal block and wiring, can prevent the corrosion/hardening from getting too bad. If you notice any discoloration or corrosion, have the cord replaced. There are also conductive pastes available, that can be applied to the wiring and terminals to help prevent corrosion. While this should always be used with aluminium wiring, it may also be applied to copper wiring.
Loose Connections
This can be caused initially by not tightening the terminals to the proper torque (usually specified by the manufacturer), or over time by heating/cooling cycles. Loose or bad connections can cause arcing and/or higher resistance. As discussed before, high resistance leads to heat.
Solution
Make sure all terminals are tightened to the manufacturers specifications. It's also a good idea to tighten the terminals after a few hours of operation. As stated before, occasional inspection can also prevent this situation.
Why didn't the breaker trip?
The heating was localized to the stove\cooktop\range, so the thermal protection in the breaker would not trip. Since stoves\cooktops\ranges draw large amounts of current during normal operation, the current during the event likely didn't go high enough to cause heating in the breaker to trigger the thermal protection. It could also be that the breaker is faulty, so it should be inspected.
Is the stove scrap?
The stove may be salvageable, though you'll want to thoroughly inspect all the wiring to insure it is not damaged. Replace any damaged wiring with equivalent parts (cooking appliances use high heat resistant wiring).
If the stove is still under warranty, you'll want to contact the manufacturer to discuss your options.
Best Answer
If you have 3 wires coming from the wall (hot hot neutral, no ground), then you hook the wires to the 3 terminals that you see. Note the odd little strap linking the neutral stud to the chassis of the range. It is "grounded" to neutral. Any fault with the neutral wire will energize the chassis at one of the hot voltages. Yikes.
If you have 4 wires coming from the wall (with ground), a little work is required, but this increases safety. First you remove that odd little strap, as you will be connecting neutral and ground separately. Then you attach the ground wire to that green terminal you just removed the strap from. That lug in your photo is a fine method to do that! Then you connect hot-neutral-hot to the 3 studs, much like the above.
The T&B lug connectors in your question are certainly good for 40A. UL would not let them label for #8 wire if they were not good for 40A. They are undoubtedly a good choice for ground wire. For "hot" wires, after torquing their screw to spec, I would wrap them with shrink tubing or electrical tape.
I agree with 'Bat that the T&B connectors are a little concerning as far as hitting the chassis or something else. That compartment doesn't apear to have a cover, and they seem awfully exposed. See how they fit (you are sure to be able to use one) and make a judgment call. As far as the ring terminals he is recommending, you will need larger ones that fit on #8 wire. They do make those, but not necessarily insulated. Then you can shrinkwrap them with shrink tubing. Just use black.
Uninsulated is good, though, as you can solder them with a nice large soldering iron, not the electronics pencil types but either the inductive gun type or the really big 16" long types. For the solder, go out of your way to find old fashioned lead solder (non-RoHS; RoHS compliant unleaded solder will crystallize over 10-20 years, it is a death sentence for any eelctronics built that way). Do not use plumbing solder, it's acid core and not lead.
On a 3-wire connection, the neutral is so important that I would not trust a ring crimp terminal alone. I would use one of the T&B lug connectors, it doesn't matter if that makes contact with chassis.