Here is how the science behind Horse Boy and Movement Method work in short:
Many autists, and people with related neuro-psychiatric conditions, end up caught in a vicious cycle between an overdeveloped (or over activated) amygdala (the brain's fear center) and an over-sensitive nervous system.
Unfortunately, many behavioral based therapies and many traditional learning environments trigger stress - i.e. the fear center of the brain - and in addition have environments full of negative sensory triggers, which again ultimately activates the fear center of the brain. If this becomes chronic it not only blocks learning but also impairs brain development, even if the basic intellect is intact - as is often the case with autists especially.
What both Horse Boy and Movement Method do, is break this vicious cycle by creating oxytocin, which is the antidote to stress hormone cortisol (the hormone that narrows focus to a threat but sadly also blocks learning). Oxytocin is also the hormone of communication; so for people who are to some degree "locked in" this oxytocin effect massively promotes connection and openness to learning.
This is what neuro-scientists call "ending the cell danger response". Neuro-scientists at the University of California San Diego have identified Horse Boy's approaches as extremely effective in ending the cell danger response in the brain.
But there is more. When you move and problem solve - which includes balancing, exploring, having to cross natural country, bouncing and so on - your cerebellum begins to produce large amounts of Purkinje cells (named for the Czech scientist who discovered these neurons) which act as a communication network within the brain, getting the brain to begin to communicate with all its component parts in a more functional way.
This bridges the gap to the pre-frontal cortex which governs logic, reason, planning, emotional regulation, delayed gratification and so on - the most effective parts of human brain function.
In addition, the right kinds of movement stimulate the vestibular system which governs balance but more importantly for us, attention - leading to long term learning rather than the kind of learning where you cram for a test but end up forgetting everything soon after.
Finally, problem solving while moving makes the brain produce a protein called BDNF - Brain Derived Neurotrophic Factor. Think of this as a kind of "Miracle Gro" for the brain. Your brain produces way more neurons when you move in this way - and this creates neuro plasticity: the literal growing of the brain to higher and higher executive function.