The way I see it there are three possibilities:
1) The creatures DO physically exist - but not in our character's world, per sae. When your character binds a creature to themself, they create a 'link' between the world of MagicDuel and the creature's world. When your character is threatened, the creature manifests in MagicDuel through the body of your character.
2) The creatures are magic that has taken on a physical form. When you bind a creature, you're actually absorbing their essence into your own magical "flow", if you will. When you fight, you call the creature or creatures essence out