Fusible link

A fusible link is actually a fuse in the form of a short piece of wire so forget about any "rigging up" of a burnt fusible link. They carry high current and a short without a fuse will result in a car fire. You can buy fuse wire at any auto parts store.
Your links will be at the starter. There will be multiple smaller wires attached to the main terminal along with the battery cable, not the purple wire on the small terminal.

Pull on each wire and the one that stretches is the bad one. Cut out the burnt section and splice in a piece of fuse wire of the same length.

What you appear to have is just a common overheated connector terminal. The wire for that terminal is likely a fuse link wire. Dull orange is one of the common colors that denote its current rating.

A little resistance develops between the mating terminals. Current flow through that resistance generates heat, and that heat causes more resistance to develop, until eventually it gets hot enough to melt the plastic housing.

The preferred way to fix this is to install new terminals on the ends of the wires, and poke them into new connectors. You can find almost any sensor connector in large books at the auto parts stores. They might have a listing for this one too.

The next alternative is to snip off a good used connector at a salvage yard, and splice the wires to your harnesses. How much of a fusible link wire you end up with is not important. All that is important is a piece of that wire is there. It is smaller in diameter to make it the weal link in the chain, and its insulation is designed to not burn or melt.

For the fastest repair, look at whether that connector is needed. Many are only needed to allow the vehicle to be assembled from parts, on the assembly line, but after that they will not be disconnected. If that is the case, the best repair to insure this does not happen again is to cut off the two wires, one on each side of the connector, splice and solder them together, then seal the splice with heat-shrink tubing. Use heat-shrink tubing with hot-melt glue inside. That will seal out moisture so the connection does not corrode.

Hi brother Wrenchtech. Let me clarify my comment about the length of the fuse link wire. The length of that section is typically about six to eight inches long, but its length is not important as far as protecting the circuit. A half inch is enough for that. The replacement fuse link wire should be the same length as the old one to insure it does not get tugged on when the engine rocks or something else causes the harness to be pulled on.

Hi Doc,
I was always told that the length of the link does matter as it changes the protection level and the amount of current it can carry. Is that wrong?

Edit: I found this

http://www.madelectrical.com/catalog/fusible-link.shtml

Yup. It is just a standard wire, but smaller in diameter. As long as current stays within the normal range, there will not be any significant voltage drop across it. You can get the same protection with a regular piece of wire of the same gauge as the old fuse link, but that insulation would melt or burn.

I only bought a fuse link wire for my own use once, quite a while ago. I do not remember what I paid for it, but it did not come with the terminal attached. It was fifteen inches long and could be cut to make two or three repairs.

I have seen cars where people installed a fuse holder and fuse to replace a fuse link wire that keeps burning open. I do not recommend that because the wire acts like a slow-blow fuse. A 20-amp fuse will burn open instantly when it sees 20 amps. A fuse link wire will tolerate more than 20 amps for a little while. That is important for things like radiator fan motors. Motors always draw higher-than-normal current until they get up to speed. A regular fuse could blow when it did not need to.