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 from yourself and let it manifest itself again. Because it's been absorbed into your "flow" it is now a part of you and obeys your will in the same manner that your hand obeys your brain's command to type or click or [insert your personal example here].
3) There's a heck of a lot more going on here than what we already know, and I can't fully understand this until I learn more about how magic functions in the world of MagicDuel.