There are different priorities in training. One of mine is spontaneity. This involves the abilities to both think and act in a quick and fluid manner, all terms that can be further qualified. Working backwards:
Fluid means smooth. Smooth can be fast or slow; what matters most is timing. Impeccable timing will neutralize many things.
Quick is acceleration, the ability to get from A to B in as short a time as possible. Fast is the speed at which you are moving when you get there. In longer performances, such as a foot race, winners may not be moving as fast as second place, but they get there first. The faster runner, not being as quick to reach speed, plays catch-up. In longer races, though, the overall speed of the winner will be decisive. It matters not how quickly you initiate movement off the line in a marathon.
Action springs from the mind; of that there is no question. Training increases neuromuscular connections, building faster, more efficient responses. How well one performs, though, is as much about the clarity of mind and will as it is about the condition of the body. Whatever we do when we train, we are training our minds how to use our body.
Notice I used "minds" in the previous sentence, the plural form of the word. We have both conscious and unconscious systems in operation. Which is in charge and how well they cooperate is key.
We spend a lot of time being aware of our conscious thoughts. Seems self-evident, but we really spend most of our time in unconscious states. It's when we tune in that we become self-conscious. This state of awareness is fantastic for thinking thoughts, analyzing information, reading this or having a conversation. It is good for directing the unconscious mind, which is where willpower comes in, but cognitive perception is always the last link in a chain that starts from physical stimuli interpreted unconsciously and then elevated to immediate attention. At that point, decisions are made, and the unconscious mind sends signals back downstream to activate physiological responses.
The unconscious mind, however, is capable of making its own decisions, whether it is to jump if we hear a rattle in the bushes, or to reach for the phone when it rings. In a sense martial art training is very Pavlovian; we see/hear/feel a stimulus and we respond reflexively. It is possible to react even before the conscious mind is activated because there are sub-brains throughout the body. These are the major ganglia, such as the solar plexus and at the tailbone. We used to mock dinosaurs that were so big they had brains at each end of their body; it turns out that isn't such a bad model after all.
There are other levels of the unconscious, however, beyond just mastery of the physical body.
There is a higher level of awareness, so refined as to be unbeknownst to many people. This is a place where things like wisdom come from, and root awareness. Nothing can happen on any level without recognition. How we move through the world, how we project ourselves, is all an image chosen on such a deep level.
When we train, we can practice mindfulness, becoming aware of every thought, move, nuance in each and every moment. Conversely, we can practice no-mindfulness, where there is no thought, or more correctly, no awareness of thought. Paradox is wonderful; there are many ways to the mountaintop. Either way, the inhibition of conscious thought is removed from the director's seat, relegated to a more appropriate role as spectator. Sometimes I've experienced it as a commentator, like a sports announcer, but such is a distraction, a ploy to pay attention not to the action but the chatter, a status once-removed. Thought may become awareness of one's thinking; the point is there are ways to capture or corral the monkey mind. What is hypnosis but fixating the mind very specifically?
Now that we've popped down the rabbit hole, how does this apply? When we learn to flow, we learn to think more quickly than other people. We recognize possibility in motion; we respond to changing circumstances before they overwhelm us. We allow intuition and feeling to operate tactically, while our conscious mind strategizes goals.