Running the mechanical part of combat blisteringly fast allows you to spend more time role-playing your character and describing the action in combat. You’re not trying to get through combat quickly because you dislike it; you’re instead trying to move past the computational elements of the game so that the story-telling can take the forefront.
By focusing on the story-telling your game can become so much more exciting. Instead of waiting for the Warlord to thumb through the PHB to look up his class features (again) you can describe how the Warlord leapt from the second story balcony, dodged a provoked opportunity attack by doing a shoulder roll, toppled the gnarled hag with a low cut to her thigh, then sprung to his feet to stare down into her rapidly dimming eyes.
Below is a list of suggestions that will help you get through the mechanical part of combat blisteringly fast. These instructions are not for the faint of heart, and though some tables may pick and choose from these suggestions to speed up some areas of a dragging combat, only the committed will achieve the supersonic pace described by this article.