The Phrase Finder has its usual detailed explanation that the current meaning is “Get to the point—leaving out unnecessary preamble.” The first use of the term was a script direction from a Hollywood director in 1929 to move the plot literally, “Cut to the case.” It was used by Helen Deutsch in an article about screen writing ran in 1944 that advised, “When in doubt, cut to the chase.” Evolution to the current meaning is exhibited by a piece in The Berkshire Evening Eagle in 1947 that announced, “Let’s cut to the chase. There will be no tax relief this year.”
The explanation ends with the entertaining observation that most “…films aimed at a young male audience involve plot devices that allow for car/boat/spacecraft chases. There is usually a token love interest before everything in sight ends up in pieces.”