Becoming aware of oneself and our surroundings is living mindfully. But one cannot completely be aware of the external world unless one is aware of their internal self, both physical and psychological.
For example, we can not truly respect and love others unless we love and respect ourselves. The practice of Yoga helps us look internally, look within and become aware of our breath, our physical body, muscle movement and coordination, our ability to balance, our ability to stay in the moment when holding the posture, our physical limitations and our willpower that helps us control our body.
This mindfulness shows us how unaware we are of our own existence, our thoughts and actions, and how we take all this for granted, as the body and mind are always working hard to keep us alive, irrespective of whether we pay heed or not.
One can not expect such awareness to come at once but it is with regular practice of Yoga that one discovers the inner-self a little more. Thus, Yoga also teaches us patience or simply urges us to not think about what is coming or what has already happened, it keeps us in the present.