To cut to something is slang for making an abrupt change.
This usage is borrowed from film and television, where a cut, among other things, is a direct transition from one piece of film to another. Literally, a piece of film would be cut, then attached to another piece, for example, to show the same scene from a different angle as recorded on a different camera, or to introduce a new scene.
The author seems to be indicating that one event proceeds directly to another event. Thus,
Watching in shock the reports of Kennedy's murder would cut to working with our government on predicting and preventing such attacks.
means that someone who was a witness to the murder went on immediately to work on preventing future such murders.
Personally, I find this phrasing awkward, though you have not presented enough context to make a judgment. Cut to indicates a transition, but not necessarily a progression; the author, on the other hand, seems to imply that one event led naturally to another. Instead of cut to, it might have been better to use precipitate, which carries both the sense of causation and of suddenness.
Watching in shock the reports of Kennedy's murder precipitated working with our government on predicting and preventing such attacks.
There are many other verbs indicating causation, like give rise to or lead to, but most do not carry a connotation of immediacy.