Eagle pose comes just in time after awkward pose! You just worked the muscles in your arms incredibly hard by keeping them up and stretching for all 3 parts and now you get to stretch out your shoulders, which were probably screaming at you by the end of the posture.
However, the wrapping of the arms is not just for stretching out shoulder muscles. By squeezing your elbows down while lifting your chest up you squeeze and massage the lymph glands that are in your armpits, which boosts your immune system. The same action is happening with the lower part of your body; the legs are wrapped and squeezed to get the same effect on the lymph glands in the lower body. All the while you are lifting your upper body and spine into a backbend again strengthening the back and abdominal muscles. This is also the 1st posture to begin balancing on 1 foot!