A lot of us drawn to yoga have very personal perspectives on yoga's power to heal. My answer isn't exhaustive, but it's mine! And it touches on more objective things that can be offered to skeptics more easily than exclusively personal experience. Yoga is a great example of a physical practice that uses physical movement for a purpose ... but the physical practice isn't (or at least doesn't have to be) the goal of the practice. How can trauma-informed yoga help survivors heal?
Positive way to spend free time. When life is hard, and even when it's not, spending your time doing something you enjoy, something that doesn't harm you, something you look forward to, is beneficial. Financial barriers can make it harder for some than others to get involved in free time activities like yoga. Even among those with adequate income who do not consider themselve trauma survivors ...minutes in life spent doing things we actually enjoy are few and far between.
Exercise (often a gentle form) with benefits. While yoga postures are not only a form of exercise ... they still are a form of exercise with physical and mental benefits. Stretching and strengthening the body are widely recognized benefits of yoga, but the mental health connection can be less obvious. One example: backbends (including those gentler than full wheel of course!) can activate the same fight-or-flight part of the brain that responds to stress - but engaging muscles (which we often do in yoga as we backbend and work to hold) activates the logical rest-digest part of the brain responsible for calming the system - so by backbending and also engaging, we are literally training our brain to respond more calmly to stress by giving the calming part of the nervous system a workout of its own. Another: Slowing down the pace of the breath, something we often do even if inadvertentaly when focusing on the breath for any purpose, signals the brain to relax or the rest-digest parasympathetic nervous system to turn on.
Mindfulness - which means noticing the present moment, the present sensations in our bodies, and our present emotions, with awareness and without judgment. Lots of activities and practices can help us focus on the present moment, and yoga is just one of them, but yoga does often emphasize mindfulness as a goal more than many other physical practices. Sometimes just being able to pause from rumination on the past or worry about the future is a huge relief you can feel. Sometimes mindfulness can lead to insight about the ways we think and react and we can begin to change those things.
Improved body-mind connection. Coordinating movement and breath can strengthen the body-mind connection which is often lost or weakened in trauma. Even less intense forms of stress can cause us to live a little more in our heads than in our bodies. Most of us believe in body language, that is, that our mindset affects how we carry our body, but the reverse is true too - how we hold our body affects our mindset. Amy Cuddy's TED talk on power poses relates to this without even specifically citing yoga.
Feeling of control. Offering options in yoga can help return one's sense of control over one's own body. Lack of control can be a major factor in trauma, so having options in a yoga practice, to choose how to move, how long to hold a posture, how deep to go literally puts some control back in a person's hands. Invitational language can be an excellent way to incorporate options into a practice.
Few people would argue that yoga alone will help people heal from trauma, but it is often one powerful tool. When I teach, I don't necessarily speak to all these benefits all the time, but I do sometimes pepper them in, both in public classes and in trauma-informed classes...without necessarily referring to trauma.
I tend to speak to the benefits that speak to me, but of course different things work for different people. There are yoga teachers who bring in the chakras or other aspects of yogic philosophy, and no doubt there are students and trauma survivors who find this helpful. I believe yoga has a lot to offer even solely as a physical and breath-oriented practice, and that's why something as simple as moving between yoga postures and breathing can help us heal.