Wordorigins.org

This phrase meaning get to the point comes from the early days of Hollywood. Originally, it literally referred to a cut from a dramatic scene to an action one (the chase). The literal sense dates to J.P. McEvoy’s 1927 novel Hollywood Girl, where it is given as a script direction:

Jannings escapes...Cut to chase.

Figurative use comes sometime later. From F. Scully’s 1955 Cross My Heart:

I am the sort who wants to “cut to the chase” As far as I’m concerned, we can read the instructions later.

(Source: Oxford English Dictionary, 2nd Edition)