Stillness is important in an asana practice because this allows for the body, mind and pranas to become quiet which is essential for healing of body, mind and pranas.
When we perform asana with the right sattvic attitude then we can balance the state of restlessness which is fed through over stimulation, desires/ambitions, too much usage of technology, distraction and agitation. In this fast-moving world, it is crucial to develop stillness in the body/mind, for its healing.
Internal stillness facilitates the cleansing of the organs, blood, skin, removal of excess doshas from the body. Particularly the vata dosha from the bones and joints. When there is tranquility within the asana the mind slows down and prana begins to flow through the nadis nourishing both the physical and astral body.
Practitioner is then ready to practice pranayama and meditation. These higher forms of yoga cultivate a state of yoga within, the state of being or abiding in oneself.