“Extra” step in counting the number of ways to have a domino tiling of a 4 by n rectangle?

enter image description here

The picture above explains a method for doing this via generating functions & finite state machines, but what I do not get is why we must record the number of dominoes used to make a state transition. Why does the number of transitions alone suffice? Why is it that we need $ mt^b$ to record a transition instead of just $ m$ ?