It is magic. The source just comes from materia. Here-

[Aerith frees his arm and his hand as she continues.] -I'll show you. [Lifting her hands into the way she has her hair pulled back, she nips into her lower lip as she carefully releases the smaller materia that she hides beneath the pink ribbon she seems to constantly wear. Then she holds it up for him to see.]

See? This is materia. And magic can be used by using these.