When the person sits on the foot, he is smashing the nerves in that region. These nerves cannot send the nerve signals to the brain in normal way. So, just for few moments the link between the brain and organ from which the signal was generated will be disrupted and the person will not have any feeling of the presence of that region. If the legs are uncrossed and if the person stands up, the nerves are not pressed and the feeling will be normal again. Generally, it is said that feet fall asleep when there is no blood supply to the feet. The nerves are playing role in causing this state. Nerves are small thread like structures that are passing all through the body. If the person on the other side in a phone call hangs up and you still have to talk to him while the phone line is cut, then it is the same situation that is happening even in the case of feet falling asleep. The brain is sending signals to the foot, but the feet will not respond to the signals and will not be able to send them back to the brain.
If the person gets up and frees his legs, the nerve compression will stop. Though the person can sense the normal feeling in the feet, a feeling of tingling might remain in the feet. The person might feel that pins or needles are pricking the legs. This feeling will remain for some time and then it slowly subsides.
As the normal nerve impulse transmission is blocked for the moment some nerves will start transmitting the signals while some do not. So, the brain is sent mixed signals which are felt by the brain as burning or tingling sensation. Moreover, when the legs are released from pressure, the blood flow becomes normal in that area and the nerves will get ample blood to send the signals back to the brain.