We are moving like 100,000 mph. And why do we not fall off going that fast
Because the rotation of the earth is very smooth and doesn't change, the centripetal acceleration we feel is very nearly constant. This means that the (small) centrifugal force from the rotation gets added to gravity to make up the "background force" we don't notice.
Taken from Physics SE.
Not only Earth is moving but also you with the rest like atmosphere, so you are in closed system. The edge of atmosphere does not fly off because it has nothing to rub against, it just stays in same speed. When you put your hand from the car and you feel the force against it, it is the force which you must make to push the air from your way. If the air would be moving at same velocity as you, your hand would not feel this and you would be able to hold outside of car even at 300km (do not do that, it is dangerous).