As it was explained in the game, Espers were once humans that got caught in the middle of the fight between the three goddesses and were transformed as they were hit with the goddesses' powers. You could say they are mutants, in a sense. As a result of their transformation, they gained the ability to use magic. Some can also fly and communicate telepathically, but probably not all of them.

Besides, I believe they were called Phantom Beasts in the original version, but I could be wrong.