Equilibrium means ‘no net force’
Earth & moon persist in their relative motion, so there are no net forces acting on their centers of mass - forces must sum to zero
Earth & moon exert gravitational attraction on each other, & so fall toward each other
Earth & moon never collide because they are moving past each other - each has inertia
Gravitational attraction is exactly sufficient to produce the centripetal acceleration required to move each body in a circular path about their common center of mass