[Etymonline](http://www.etymonline.com/index.php?term=bust) 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 involved, but "bu[r]sting in" to places where criminal activity is going on is as good a guess as any.