We can consider the action of an NPN BJT using band diagrams. With no applied voltages, we see the following shape:
Clearly, no current can flow, as the "hump" in the middle stops current in both directions. Now, if we apply voltages as we did above we get the following result:
The EBJ potential barrier lowers, and some electrons can drift across into the base, whereupon they are swept up by the electric field from the collector, and "fall" down the potential hill into the collector. A few holes also diffuse from base to emitter.