In Monster's Ball (2001), why did they cut only part of the jeans pant of a criminal like this?
Because that's where they attach the electrode when someone is executed on an electric chair. From the Wikia page about electric chairs [Warning: A Bit Graphic]:
In order to go to the chair, the prisoner must have their head freshly shaved only minutes before the execution. They also have to have cotton stuffed up their anus, in order to keep them from messing themselves, since the electricity takes away their control of their bowels. Next, they have to have their legs shaved, so an electrode can be placed on their right leg, and all facial hair is shaved. They have a slit cut in the leg of their pants so the electrode can make contact with their bare skin. Once that is finished, they are led to the chair.
Sometimes, instead of a slit, the pant leg is rolled up or, like in this particular case, cut off altogether. The idea behind using both the head and calf electrodes is for the current to pass through the body, hitting all the major organs along the way.