For one, I think many of us would feel good to see such a person subjected to the worst kinds of suffering until he begs for mercy. — Agustino
It would be a very sad day for humanity indeed if, as you say, a majority of people would feel good at seeing the suffering of another human being. It reminds one, of days in medieval times on which criminals were disemboweled before a crowd of commoners, all shrieking with delight while drinking in the ghastly spectacle.
You've failed to understand the mental state of people who end up behaving the way you describe in the OP. People of this sort are termed "psychopaths'', are born with no sense of empathy, kindness, etc. and therefore cannot help but act in these ways. Therefore, torturing them would be the equivalent of torturing a three year old kid who just ripped off the pages of your favourite book, without of course, knowing the value of the book.
A more rational reason is so that other criminals who intend to commit similar crimes see what will happen to them and repent sooner rather than later. And the final reason I have is that such a punishment ensures that justice is adequately done - which is required for people to have faith in the justice system. — Agustino
The last two reasons you cite are, in short, deterrence and increment of public faith in the justice system. To deal with the first,observe that an exactly equal amount, if not greater, amount of deterrence would be the result if the criminal was locked away for life in prison. Remember that prison is by no means a nice place, and many criminals would much rather choose a short interval of sharp torture than an eternity of long, drawn out torture and molestation by the not quite so friendly inmates of modern prisons.
As for the second reason, I don't see how people would end up having faith in a brutal criminal justice system which relishes torturing people. What,in reality, would happen is quite the opposite. The public would see this uncivil justice system itself as the enemy, and thus would no longer feel comfortable handing over their squabbles over to receive what they would,not unreasonably, see as warped judgment. Instead then, they would start "settling'' their disputes on their own, which would lead to mafias, clan wars and later, the disintegration of the entire fabric of society.
I hope this convinces you.