Straight-ahead speed has been broken down by conditioning specialists like Vern Gambetta as starting speed, acceleration, top speed, deceleration and cooperative speed.
Starting speed is largely a response to some stimulus that involves a series of cognitive processes. For example, you are covering a striker. Their midfielder sees your striker, looks down, and strikes the ball. Who gets to the ball first?
In order to intercept this pass, you have to make a number of decisions quickly: What space is the striker moving to? Is the ball played to feet or space? How fast are they moving? What is your speed? When do you have to start running to beat them to the ball? How about the pace and spin on the pass? How do you time your response to the pass in relation to the striker's speed? Is the ball on the ground or in the air?
If it's in the air, you have to plot out the flight of the ball and determine where on the field and your body (head, chest, foot, etc) you plan to first contact the ball, then add in the opponents skills and speed.
Then factor in what you will do with the ball. Control? Head the ball? One-touch? Two-touch? Shot? Clear? To whom/where?
All this and more is done in fractions of a second, every time something changes with the ball. These are all part of those mental features that end up being speed of thought and reactions. These arent reflexes; these are reactions not the same thing.
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A reflex, like the knee-jerk reflex, doesnt involve the brain. A reaction does, because there is input from many places to process and interpret and then decide on a coordinated response.
How does one get better at this? Deliberate practice and repetition. Some say the real difference between the elite and the not-so-elite is that the elite players have practiced skills so much that the execution of the skills is second nature, performed on a subconscious level, so to speak, that lets the conscious part of the brain focus on tactics, not skill.
But the running part can be improved. This too is mental because improvement in running speed is largely changing how you run the skill. In soccer, improvement in top-end sprint speed is not all that important.
Why? Look at 100-meter sprinters. These runners dont reach top speed until the middle third of the race; it takes 30 meters to reach top speed. In soccer, full, all-out sprints (i.e. over 30 meters) are pretty rare. Most runs are of 30 meters or less.
What that means is that the time spent teaching one to increase top-end sprint speed might be time better spent on other lessons, like the first 30 meters where the player is reacting and accelerating, but never quite reaching top speed. Thus, the initial first steps are important.