A few reasons actually:
"Yoga with shoes is like playing dodgeball with flippers."
Yoga is about creating awareness within your body, connectivity to those around you and cleanliness of mind and body.
1) Yoga is meant to remove distractions and excess from your range of awareness. You are also trying to create a connection with everything around you and experience the sensations of the body.
Tip: Try grounding through both of your feet and lifting just the toes, which you can't do with shoes. It strengthens the arches and creates the connection to the earth below.
2) We practice the postures to keep the body supple and open. If, for our entire lives, we always have shoes on, there is never an opportunity to allow the bones and muscles of our feet to be free. As our bodies get older and our feet stay closed, our walking gate turns to a shuffle and our stability wavers.
Tip: Try interlacing your fingers between your toes to create some space.
3) Yoga is about purity of body and mind. And taking your shoes off is sort of the yogic equivalent to a sign I remember from swimming somewhere. "We don't swim in your toilette so don't pee in our pool." A bit crass but we try and keep everything clean for you so that mentality of constant cleanliness is ever present. The experience of the studio should be pristine, both in spirit and practice, so we leave our shoes at the door, along with the "mind clutter".