Something to add to this that most non-neuroscientists don't realize is that the spinal cord is not just a bundle of wires that connect the brain to muscles. The spinal cord is a complex circuit that contains "central pattern generators" (CPG) that can produce rhythmic movement. When activated, these networks can be pushed into a dynamic state where they repeat the same movement repeatedly, e.g. walking. For more, look up the work by Sten Grillner on stingray CPGs.

One approximate way of looking at this is that the brain's motor cortex sends a "go" signal to the spinal cords CPGs and these start generating the signals to the muscles to walk. You can also think of motor control as hierarchical where as you go up the hierarchy. The bottom of the hierarchy are the muscles/actuators, then the neurons that stimulate the muscles, then the circuits in the spinal cord that have these CPGs and other "primitives", then primary motor cortex and further up the cortical hierarchy. At the higher levels of the cortical hierarchy are representations of whole movements, like moving your hand to mouth. As you descend the hierarchy, the neural signals control gradually more details components of the movement. For more, look up the work by Michael Graziano.