The Urban Dictionary attributes this tem to the famous radio program that aired from 1936 until 1957.  “The sound effects of police sirens, tommyguns, and screeching tires that opened the show were dramatic and exciting…(and) inspired the expression ‘coming on like gangbusters’.”  The usage has evolved to describe things that are exciting, successful, and intense, and the ‘coming on’ part is often omitted.  As an example, you might hear someone say, “That business idea has taken off like gangbusters.”