When we study yoga we start with Ahimsha because it relates to all of the other yamas.
We practice Asteya (non-stealing) on the mat, not stealing from or compromising one part of the body or quality of the breath to express an asana - which would do harm. We might then think about parts of our life out of balance, where might we be stealing from? What might we be stealing in our relationships?
Asteya is a big one for me personally as I have to be careful to not 'steal' too much home life for teaching, practicing and promoting yoga for wellbeing in our community - before I take something else on I reflect on asteya. Overwork is one of my Samskaras - one of the grooves of habit I can quickly fall into and the reason I decided to dedicate myself to yoga. Asteya helps me to keep that my biggest samskara in check.
We practice Satya (honesty) on our mat, being honest about where we can go with steadiness and ease and we take this off the mat too. Honesty in our actions, words and thoughts to ourselves and to others - because dishonesty would cause harm, but we also want to reflect back on the principle of ahimsha so that we don't use satya in a way that will cause harm.
We practice Aparigraha (greedlessness, or non-grasping) learning to take only what is truly necessary and no more and to let it go when we no longer need it. This is a big one on the mat, we want to be motivated to explore tha asana and the body and mind but we need to balance this with non-grasping, are we reaching too far, are we obsessing about the next asana? When we focus on 'more' we can sacrifice what is here - in this moment - which is the only place that life is. Grasping and attachment cause suffering because everything is temporary and everything must pass, if we are so attached we will suffer when that which we are attached to is no more.
Brahmacharya sometimes interpreted as celibacy, but more often as wise use of energy and fidelity. Wise of energy is of course particularly apt on the mat, we don't want to deplete our energy early in the practice, nor do we want to deplete our energy for the day and this can help guide the structure of our days off the mat too. This also helps me to ensure I'm not depleting my energy in yoga outreach work depleting my energy for my practice, teaching or homelife.
All of this has the potential to help us to grow and change. 
This is why we come to the mat even (especially) when life is hardest, because we know that the mat is a place of reflection, strength, flexibility and growth.
