In olden times, after a pitched battle, soldiers of the victorious army would drift around the battlefield executing any enemy soldiers - wounded perhaps - still remaining...ie showing no mercy.
Since the Geneva Convention, things have been a little more civilised...that is, such soldiers or any who simply surrendered are now supposed to be taken prisoner and reasonably looked after.
When the words, "He takes no prisoners" are used nowadays, they just mean that this person is utterly ruthless and means to get what he wants regardless of its effect on anyone else he deals with.