The buildup of charge in a thundercloud causes a voltage between the ground and the cloud. When this voltage is great enough dielectric breakdown of air occurs. This means that air loses its insulating properties and becomes conductive. In this state the air has free electrons that will move due to the electric field between the cloud and the ground.
Electrons themselves don't travel the distance between the cloud and the ground in split seconds. However, all the free electrons in the lightning's path will be moving in the same direction, on average. This is called the drift velocity and it's quite slow, though I don't have an example of drift velocity values in air where dielectric breakdown has occurred. In a copper wire the drift velocity is of the order 10$^{-5}\frac{m}{s}$, the exact value depending on the applied field/voltage and the dimensions of the wire. The high electric current is not a few electrons moving very fast, but a large amount of electrons slowly drifting in the same direction.