Hmmm, my reply melted in space, I'll try again.

I mounted a starter solenoid on my right fender and it works well.

Your solenoid could have died a natural death with age but it could also be your starter is drawing too much and fried the relay. If you have access to a suitable amp gauge you could install it between the solenoid and the starter and determine the amp draw of the starter, if it's out of spec you know you have a starter overhaul in your future, if it is in spec you know the solenoid was indeed expired. One of the most common ways a solenoid expires is from the studs rotating while tightening the cable nuts. This can cause the contacts to misalign and stay connected when the key is not in the start position. It is also the most common reason for starter run-on after starting the engine. It could also result in just a small portion of the contacts actually contacting and causing a thermal runaway type situation and melting the solenoid. As for the engine continuing to run when you turned off the key, with a stock wired car there is a wire with +12 volts that runs to the coil part of the ignition system when the key is in the start position, bypassing the ballast resistor in a points system. When the key is in the run position that wire is taken out of the circuit and another wire supplies the voltage to run the ignition circuit. +12 volts for an HEI and about +9 volts for a points style distributor. If the solenoid is faulty and is staying in contact after the start it will continue to supply +12 volts and the car will stay running even when the ignition switch is in the off position.

As for how it works and why it has a benefit, the advantages are many.
If the solenoid is mounted on the right fender near the battery the only cable that is live when not in the start mode, is the 12 or so inches from the battery to the solenoid. The portion of the cable that runs to the starter is dead in the run mode.
This eliminates live cables and small wires from the hot area of the starter and exhaust manifolds or headers.
The wire from the ignition switch to the S terminal of the starter is re-routed to the solenoid and a buss bar or jumper wire is put between the S terminal on the starter to the battery cable terminal on the starter. This eliminates the wire from the switch running through the tube and down to the starter, keeping it cooler and supplying proper voltage to the s terminal of the starter and thus the starter mounted solenoid. This can help eliminate hot start problems.
Some accessories are mounted on the BATT terminal of the starter these are moved to the Batt stud on the remote solenoid keeping them away from the heat of the engine and the exhaust and reducing voltage loss due to increased resistance in the wires caused by heat.
The wire mounted on the sarter R terminal that runs to the coil is mounted on the remote solenoid keeping it away from heat and also a full battery voltage on startup.

When done you should have the cable from the solenoid running to the starter and no other wires, just the cable.

The remote solenoid is a convenient place to wire in accessories for full voltage and also a convenient place to instal a remote starter switch when doing maintenance.

In my opinion it's just a better way to wire in the starter and related circuits. Some like to have their cars appear stock, the relay can be hidden under the fender. Some like there car to be stock, for them this is not an option.

The biggest advantage is lower risk of electrical fire and full voltage to starter.

Mine has an HEI and a relay on the firewall supplying it full system voltage via a terminal block also mounted on the firewall. Charging voltage is supplied to the solenoid mounted on the fender also via the firewall terminal block. The firewall terminal block is fed from the alternator and the voltage sense wire is attached there. A second terminal block on the rad support supplies voltage to the five circuit breakers and relays feeding the rad fans, high and low lights and horn. Everything fore of the firewall gets full 14.5 volts when running and full battery voltage when starting or in assy mode. Before the changes I was getting 10.5 volts due to loss from resistance in the wires running from the alternator to the dash and back.

I hope you made it to the show.