HIIT (High Intensity Interval Training) is commonly misused while training for speed. Most people are slow because they are out of shape. HIIT gets people into shape quickly, so it naturally improves the average person's speed.
However, elite athletes have programs specifically tailored to increasing their speed, including the use of resistance training to help improve their explosiveness.
The biggest thing you can do to increase your speed? Stretch daily! Flexibility is closely related to speed.