If snake has venom in its teeth, how come it does not die when it bite and eat its prey?

Yes, if a venomous snake bites itself and injects a considerable amount of venom, it could quite easily die. Remember that the venom itself never comes into contact with the snake's bloodstream - it is stored in venom glands behind the snakes eyes and injected through the fangs into the prey item, and then the prey item is swallowed into the stomach, never touching the blood.