They can model many problems that involve state transitions nicely. For example, imagine you have a pulse going through a wall made up of several layers. The signal is attenuated (reduced) by travelling through each material, and partially reflected/transmitted at each boundary. What is the total energy that you receive back at the transceiver? If you try doing the sum you get a very difficult problem. But you can rephrase the problem in terms of a directed weighted graph and make things much easier:
Have vertices representing the start node, each intermediate layer (which gets two: one per direction of travel), and one outgoing node for the signals that are simply lost on the other side. The edges are weighted according to how much the signal is attenuated when going from one state to the other (material attenuation and reflection/transmission combined). Now the result you want can be calculated by performing a simple calculation on the weighted adjacency matrix.
The point is that graphs model "I have stuff, and it's connected to/interacting with other stuff". This is an absurdly general concept, and so applications of graph theory will pop up in all kinds of neat places.