The mechanism within our body linking mechanically sensitive neurons in muscles, tendons and joints called proprioceptors to the central nervous system which enables to define milestones in a child’s growth- crawling, standing, walking, climbing stairs, running are nothing but witnessing development of situational body awareness. The 'sport' of climbing only taps into this and enhances it, adding massive benefits towards a child's development. Basically, advanced proprioception allows you to learn better and do more.
Other reasons are-Climbing helps building confidence, enhances problem solving skills, increases alertness, self confidence and cultivate ability to convert failure to learning lesson.
The bonus reason- Climbing is now a medal event in Olympics. Who knows your kid might be the Olympic medalist we are looking for.