1998S70 Volvo jump started it and put the jump

Did you replace the battery as well? If both cables are badly melted it is likely the battery has suffered damage as well. Do you have access to a Multimeter? Set it to Volts DC. With the engine OFF put the red pin to the red positive battery terminal and the black pin to the black negative terminal. If the battery is good you should get a reading of around 12.6 volts or higher. Anything below 9 volts and the battery is probably worn out or damaged.
You will also need to find out what caused the cables to melt. There is most likely a 'Resistance' issue somewhere in the electrics. This is where the voltage is restricted and cannot flow correctly. This generates heat and is possibly the reason the cables melted. Have you 'Jump Started' the vehicle recently? Connecting jumper cables the wrong way around can have a similar effect.

In this situation I would check the starter motor solenoid, you are lucky the ECU did not fry cause it would cost more than the car is worth. Starter motor is not that expensive, also check all main relays and fuses. Check the wiring as well. If all fine the ignition coil and ignition barrell would be last. If it is possible plug a computer and run diagnostics too.

I have never seen a fusible link on a Volvo.

if the battery is ok, you should check the starter and ignition, they may be cause your car won't start.
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One of those secondaries will be the feeder for the ignition switch. The other may be for the fuse box, or may be for something else that is powered at all times-like the headlight switch. Here's how to clean it up:
First take off the negative battery cable-negative first to avoid sparking. Now you can pull off the positive. Most new positive battery cables only come with one "pigtail" or secondary insulated wire. But if you tell the parts person you have a Volvo and the year, they may have a cable specifically for your car with two secondaries. Or you could try the dealer for the right cable.
Barring that, if you just want to fix your old cable without buying a new one, you could take the stripped wire and crimp a new eyelet connector on the end and wrap the bare portion with several wraps of electrical tape. Now put the big cable terminal on the battery post, take off the nut on the tightener bolt, put the eyelet connector over the bolt, and tighten the nut down on the bolt. Now both secondary wires are attached? Put the negative post on and tighten it down. To avoid corrosion at the posts you can smear a little petroleum jelly (vaseline) on the post and cable connections, if you want.

Check battery cables are correct ! Prime suspect here ! Alternator may be damaged and shorting .. Don't drive any car that will melt jumper cables ! Something very basic wrong here.

A weak battery will make the "clicking" sound you hear, also bad connections, loose battery cables or worn battery cables. Jump start the car to see if it cranks, if it does, charge or replace battery.

THE JUMPER CABLES WERE OF A TOO LIGHT OF GUAGE WIRE. THERE SHOULD BE A BATTERY FUSE LOCATED IN THE FUSE BOX IN THE DRIVERS SIDE ENGINE COMPARTMENT.

with a new starter, this sounds like a bad battery cable or terminal end. the easier way to figure this out w/out checking the voltage drop across a whole battery cable is to affix the jumper cables strategically to find the bad one. Put jumper (+) on batt (+) and jumper (-) directly onto a piece of unpainted engine metal and not in the way of moving parts. If it starts this way then you've got a bad ground cable. If it didn't start, put jumper (+) on the hot post of the starter solenoid (there are 2 posts, only touch the one w/ the fatter cable going to it) or on a remote jump point if equipped, and put jumper (-) to batt (-). If it now starts you need a positive battery cable, or just a battery-to-starter cable (if separate)

When you put the new starter in, you didn't put one of the cables back on the post. It probably grounded itself enough to start and run, then moved by vibration, or a pothole, or gravity, and killed the motor. As for the arcing battery.... are you sure you didn't attach the jumper cables to the wrong posts?