Read Maty Ezraty's response:
Transitions from pose to pose are often neglected in yoga classes. Students tend to rush, forgetting to take time to get in and out of postures. It is during these quick transitions that students can become inattentive and hurt themselves.
It is also in the transitions, or the pauses, that we can learn the most about yoga. These pauses create times to reflect and consider the effect of each pose. This is one reason why your teachers have you pause before coming up from Savasana. Staying in touch with the calm and focus of a deep Savasana is much easier if you do not jump out of the pose. Take time to stay with that feeling of relaxation as you slowly pause on the way back to "normal" life. This is the real yoga-bringing your experience from the mat into the rest of your life.
As far as I am aware, the reason for coming out on the right side is that it is away from the heart. It is also better if students do not bump into each other, which is less likely if they all turn the same way. I am not sure there is anything magical about coming out only on the right side, although I do generally stick to that practice.
Maty Ezraty has been teaching and practicing yoga since 1985, and she founded the Yoga Works schools in Santa Monica, California. Since the sale of the school in 2003, she has lived in Hawaii with her husband, Chuck Miller. Both senior Ashtanga teachers, they lead workshops, teacher trainings, and retreats worldwide. For more information, visit http://www.chuckandmaty.com.