This context comes from the movie "Midnight Run". It's a scene where one of the characters uncuffs the other.
"Thanks, 'cause it's starting to cut into my wrists. Thank you. It gets the skin after a while."
- to hit, strike, or wound: The bullet got him in the leg.(Random House Kernerman Webster's)
d. To annoy or irritate: What got me was his utter lack of initiative.(American Heritage® Dictionary of the English Language)
The second example seems to refer to a person's emotions although skin can also be irritated.