JoshP's answer is good and I'd like to add more detail about the relay action. When amps switch on without the anti-pop relays incorporated, it's a lottery how the push-pull output transistors come into correct working - one transistor may become switched-on several milli-seconds before the other and this will inject a pulse into the speakers which may damage them or just prove annoying.
The relay may be used for other things but basically it is supported by a small circuit that monitors the output from the power transistors to check that they have stabilized following application of power. The relay contact then connects one or both speaker wires to the amplifier in the knowledge that there won't be a big popping sound.
Because the relays are magnetically driven switches there is no significant contact resistance to add colour to the signal reaching the speaker. So on power up and soon after you hear the relay click. When you power down there will be a small delay while the relay coil circuit (the thing that creates the magnetism to close the relay contact) discharges its energy thus a small click after power is removed.
I would say there's a fair chance that because the relay can directly disengage the speaker from the power transistors that this may be the culprit in your failed amp. Alternatively it may be doing its job and preventing a shorted (or partially shorted) transistor doing the speaker damage.
Output capacitors can also fail and this will cause the same problem - a large amount of dc onto the speaker - the relay protects against that too. Alternatively the AC power supply section in the amp may have died and nothing will work unitl this section is mended.