This article is from the alt.usage.english FAQ, by Mark Israel email@example.com with numerous contributions by others.
On its AOL message board, Merriam-Webster Editorial Department
writes: "The phrase 'cut to the chase' developed from cinema
terminology, where it referred to the act of switching from a less
action-packed scene to a more exciting sequence -- typically a chase
scene -- in order to draw the audience's attention back to the
screen. Within the past fifteen years or so, 'cut to the chase' has
come to be used outside of the film industry with the figurative
meaning of 'get to the point.'"
Jesse Sheidlower adds: "The literal use -- as a director's
instruction to go to a chase scene -- is quite old. A 1929 novel
about Hollywood has 'Jannings escapes....Cut to chase', for example.
The figurative use, which is now quite common, is fairly recent; it
seems to date only from the early 1980s."