How did the word 'busted' morph into a synonym for 'caught'?
Busted, down on Bourbon Street, Set up, like a bowling pin. Knocked down, it gets to wearin' thin They just won't let you be....
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Green's Slang Dictionary has bust meaning a break in or a raid dating from 1865, but the earliest reference to a police raid is from 1938. Later the word came to mean any arrest or criminal charge. This usage probably derives from burst meaning a burglary, which it dates from 1834.
I can recall Philip K. Dick using the word burst as a typically Dickian substitution for bust in one of his novels (sadly I forget which one), which would have required commendable self-discipline in the days before word processors.
Etymonline tends to back up MrHen's guess:
variant of burst, 1764, Amer.Eng. The verb sense of [...] "arrest" is from 1953 (earlier "to raid" from Prohibition).
Unfortunately this doesn't give us a lot of insight into how that meaning evolved, but "bu[r]sting in" to places where criminal activity is going on is as good a guess as any.