The short story is: a small base current controls a larger collector current. The short explanation is that it does this by changing the potential barrier in the crystalline lattice of the semiconductor material. The lowerering of this potential barrier is what allows the increase in current carriers.
I was able to google this link to a slide presentation. It looks like it is fairly thorough coverage of many of the models that are used to study different aspects of the BJT transistors. It includes treatment of the Re model.