For many people, Pigeon is their favourite hip opening pose. And for good reason. You stretch the side of the hip and buttock of the front leg (hip rotators) and the front of the hip of the back leg (hip flexors). So when you practise two sides you have 'opened' a large area of your hips. Along with creating space in the hips, this pose is also very beneficial for the lower back. You will feel more freedom in your movements in daily life.
When you relax your jaw, your face and the hips in this pose, and surrender to the sensations of the hips releasing tension, something special happens. Time stands still, the mind calms down. There is space for emotions to surface and if you manage to stay present and just observe what happens, they'll disperse, leaving you soft and more relaxed than before. This may also have to do with the idea that the hips store lots of emotions.
Pigeon pose can be practised as part of a flow, only staying for a few breaths. Also, you can hold this pose for up to 5 minutes as a Yin pose (in Yin yoga, this is called 'Sleeping Swan'), giving space to everything that wants to arise while you practise staying present. This way you really give the hips some time to open up.
Have a look at my step-by-step instruction for Pigeon pose.