First disconnect the main negative battery terminal under the hood. This disables your entire electrical system.
Cut the wire slightly beyond the burnt part to make sure you make a cut in "clean" cable, if it's still long enough you can use it. Inspect it carefully. If anyother part doesn't look right, replace the whole thing. It's not expensive. Get a new crimp eye terminal to connect the wire to the car. Sand down to bare metal to get a new ground connection.
Make sure all your other power, ground, speaker, rca, remote wire connections are tight, not burnt or worn. Physically touch them and try to tightent them down a little more. They don't have to be gorrilla tight, but tight enough that they aren't comming loose without a tool applied to them.
Now take your meter and set it to continuity. Put one probe on the negative terminal on the amp, the other probe on any exposed metal surface or exposed screw. You may need to scratch the screw or find a place where the paint is scratched off the car to get a good contact. Your meter should read near zero or beep at you when you have continuity. That means your ground wire is passing current from your amp thru the metal chassis of the car.
Now reconnect the negative terminal on the battery under the hood. Now set your meter to DC voltage. Now test between the negative and postive terminals on the amp. It should read around 12 volts or slightly above.
Turn on your engine and turn on your stereo. Test the same between the remote terminal and negative terminal. It should read the same.
Your stereo should work normally.
