Silent movies often featured chase scenes, which were often the most exciting/interesting scenes. So, ‘cut to the chase’ came to mean “get to the important part.”

Not to add to the discourse, but… It’s ‘toe the line,’ not ‘tow the line.’ It’s ‘free rein,’ and not ‘free reign.’ It’s ‘eke out,’ not ‘eek out.’ It’s ‘sleight of hand,’ not ‘slight of hand.’ It's ‘Anchors aweigh,’ and not ‘Anchor's away!’