Here is a great article explaining some of the benefits of exercise for humans, and specifically children.
Since the late 1990s, research has revealed that aerobic exercise
• boosts levels of brain-derived neurotrophic factor (BDNF), a substance essential for the growth of brain cells
• stimulates neurogenesis—the birth of new neurons
• mobilizes the expression of genes that are believed to enhance brain plasticity—i.e., the ability of the brain to change its neural pathways and
• prevents brain tissue loss in older adults
Fit children also showed distinctive patterns of brain activity. For example, when school children were asked to view some images of animals and make quick judgments about them ("Is it a cat or a dog?"), physically fit kids had faster reaction times, and their brains showed evidence of more extensive processing during the task (Hillman et al 2005).
In one recent experiment, kids who ordinarily performed poorly on attention tasks improved their accuracy when tested shortly after "moderate acute exercise" -- 20 minutes of walking on a treadmill (Drollette et al 2014). A similar experiment found that a 20 minute session of walking boosted children's performance on tests of reading, spelling, and arithmetic (Hillman et al 2009a).
And what about ADHD? When researchers tested the effects of short-term exercise on kids diagnosed with ADHD, they found that aerobic activity gave these kids a special boost, altering their brain activity in ways that might enhance self-discipline (Pontifex et al 2013).