Student or Learner
If busted means stopped working, the radio was busted. how come "bust ones butt" means work hard?
In AmE, "busted" has many meanings, but none of them are "stopped working."
The most common meanings of "busted" are 1) caught doing something wrong, and 2) broken.
So when someone says "I busted my butt" it means the same as "I broke my back trying to please my boss." (The person is using "broke" in the figurative sense, of course, meaning that he struggled and worked above and beyond what was expected of him.)
As for definition (1) above, "busted" ranges from actual criminal activity ("I was busted selling drugs to an undercover cop") to behavior officially prohibited by certain rules ("My boss busted me for Googling porn sites on my computer at work").