Have you ever played football?
It's all about focus/concentration. Having the ball at our feet, we focus much on the ball, keeping it always under control with the right touch/weight while moving rapidly, always making sure our foot are always near it, when we turn here and there, running with the ball. In order to always be near with the ball, we instinctively push all our body forward and increase our paces (the moving of both foot right and left towards the ball in heightened intensity)... it's kind of like chasing the ball when we burst forward with it, in order to always be in control of the ball. It's important to train and sharpen always our football touches.. that's the key to be that fast on-the-ball.
It's very different off-the-ball, there's nothing to focuses on, or aims to target.. plus there are a lot of "distractions" eg. players moving around, noises in games, etc. Well there are those players who are fast without the ball than with, the runners when they run, they focuses on their legs or/and the spaces they're running into, often best at straight line. And then there are those who are that good on- and off- the ball.