The click of death normally happens when the hard drive heads cannot tell where the tracks required are on the platter(s). This causes the heads to keep moving until the actuator arm that holds the heads hit a physical stop. The click of death sound that you hear is the sound made when the actuator arm hits the stop. The stop is there to stop the reader heads from leaving the platter(s).
These two video’s are of hard drives with the dreaded click of death.
The click of death is normally caused by a fault with the heads and sometimes (although much less often) by a fault on the hard drive circuit board. In both cases the safest option to give the best chance of getting any data from the clicking hard drive is to use a data recovery service. They have the clean rooms and specialist equipment required to take the platters out of the faulty drive and possibly recover the data.
If a data recovery service is not an option for you, or you like to tinker with these sorts of things, you can try and get a working identical drive and try replacing the circuit board on the faulty hard drive with the circuit board from the working hard drive. If that does not work the next step would be to try and move the platter(s) from the defective drive into the working hard drive. I have not done either of these steps and would personally just use a data recovery service if the data was that important to warrant the time this would take and cost of an identical hard drive. In saying that, a number of people have been successful in resolving a number of different hard drive problems by replacing the circuit board. However, I would say that very few people have been successful in swapping platters, as the chances of contaminating the platters without a proper clean room are very high. Try either of these solutions at your own risk!