Why does "I got 'busted' for smoking weed" mean 'caught'?

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.

I would have thought that it has its origins in the law enforcement term '[To carry out a drug] bust'. Although it is often used interchangeably with the term raid, I would suggest that the physical act of gaining forced entry by "busting the door down" led to its becoming known as such.

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.