When a bird is flying or when a copter is hovering or a plane is flying what force is making them rotate with the earth beneath them?
They aren't. To first order, airborne objects are moving with the Earth because they were moving when on the ground. However, what they retain is their angular speed, and since the circumference increases with altitude, an object that doesn't accelerate appropriately will appear from the ground to fall behind to the West (the Earth is rotating to the East). This is a well-known phenomenon whose magnitude is latitude-dependent, falling to zero at the poles.
When something is flying through the air, it keeps moving along with the Earth because it was already moving that way to begin with. Remember that according to Newton's First Law, things will keep moving in the same direction unless an outside force acts on them. So when a bird, plane, or helicopter is on the ground, they are rotating along with the Earth. In the rotating reference frame of the Earth, they are stationary. From an external frame which sees the Earth as rotating, the object will be rotating in the same way. Now, when a plane takes off, it is changing its motion relative to the ground. In that reference frame, it begins moving. However, from our external reference frame, it was already moving. So its just changing its already non-zero velocity.