When we use this expression, we are usually telling someone else to hurry up and get to the point. We may also use it about ourselves to express that we are going to tell you the most important thing, first.
Example: Let’s say that there was a fire in my office - a big fire. There three fire trucks as well as news cameras reporting on the situation. I got out of the building, as did everyone else. When I call my wife, I might begin the conversation like this: “Hi, honey. There was a big fire at my office today, but let me CUT TO THE CHASE. Everyone is fine.” Then, I’ll proceed to tell her some of the details.
When we cut to chase, we are telling somebody the most important thing - the thing they most want to know about a situation.
One of the reasons Furious 7 is so much fun is that it consists of almost ALL action scenes. Lots of car chases. This is where the phrase came from - making movies. When you go to see an action movie, you often don’t want to be bored by all the talking scenes. You want them to literally cut to the chase scene!
The other reason I wanted to post this is because I love this picture. I wish somebody would make a movie out of this.