When we are lying down the requirement of oxygen in the muscles of the legs and arms is reduced. The heart adjusts accordingly and slows down the rate of pumping blood to the entire body along with the arms and legs.
The amount of blood being pumped by the heart is controlled by the constriction of the veins and arteries. This is a reflex reaction. On standing up suddenly the reflex reaction is not able to change the flow of blood immediately. This reduces the amount of blood being sent to the brain and is called postural or orthostatic hypotension.
Usually this is corrected fast, so your dizziness is only momentary.
I believe the answer posted above needs some clarification. It is true that when we suddenly get up from a resting or lying down position there is a sudden rush of blood in lower limbs and less blood in upper parts of the body including the brain. And the dizziness we may sometimes feel when we stand up too fast, is because of the reduced blood supply to the brain. But I would like to clarify that this is more because of action of gravity rather than because of heart pumping more blood to lower limbs as compared to brain.
When we suddenly stand up from lying down position, more blood tends to collect in lower limbs due to gravitational pressure, and the return flow of blood to the heart is reduced. This causes a short term reduction of blood pumped by the heart. This in turn leads to reduced blood supply to brain. However the body quickly adjusts to the changed situation by increasing blood pressure so that adequate supply of blood to brain is restored. In this connection it is worthwhile noting that general blood pressure level while lying down is lower than that while in sitting up or standing position. This is because of the need to overcome greater gravitational pressure in the sitting up and standing position.