A fantastic question which is a bit of a head-scratcher at first, but can be explained when understanding that momentum is conserved.
First, a little experiment. Stand up and jump. Now, did the wall of your house splatter into you at 1000mph (exact speed obviosuly depends where you are in the world). No? Good. The reason is that although the Earth is moving and everything on the Earth is moving with it and has momentum. Imagine an observer in a spaceship far enough away from the Earth so it's not moving and they're watching the Earth rotate before them. To their point of view your jump doesn't actually follow a straight line, it follows a parabola. During your little jump your initial takeoff spot is about quarter of a mile away, but the Earth has moved exactly the same amount as well because you were already moving at the same speed as the Earth when you took off.
It's the same idea as if you are in a car travelling down the road at 60mph. If you toss something up in the air, does it shatter through the back window? No, because it it travelling at the same speed as the car and momentum is conserved.
When it comes to planes, it's important to remember that when it's sitting on the ground it has momentum because it's rotating with the Earth. When it takes off we know that momentum has to be conserved, allowing it to fly to its destination without the Earth rotating away underneath it. The atmosphere also (pretty much) rotates with the Earth again allowing the plane to fly.
A plane flying in the same direction that the Earth rotates may actually get to its destination quicker because the atmosphere is rotating in the same direction.
This is about as far as my physics goes before my head hurts!