I'm terribly sorry you have been left hanging. I have not been on the Forum for a long time; my computer couldn't get signed in for some reason.
There should be a disconnect switch near the condenser unit out side. Turn it off and return to the compressor. You will need to pop a spring clamp off the plastic cover on the compressor termianls and remove it.
Go to an electrical or A/C supply house and get new ends to go on the wires (take an old one with you to be sure of what you get). Cut the burned portion of the wires off, strip them about 1/2", crimp the new ends on and reconnect. Make sure the wires go back where they came from (maybe do one at a time). The click you have been hearing is the 24v solenoid in the contactor, which appears to be working fine. Close it all up and turn the power back on, adjust the thermostat and give her the ol' smoke test (if it works and there is no smoke comming from it, then it has passed the test).
Please post back and let us know how it worked out for you.