In a circuit, an electric field is created. This electric field forces electrons to move. As current must remain constant (since no charge build up is observed), the electric field must be strongest in the materials that are the least conductive (or most resistive).
As this electric field moves the electrons, they gain kinetic energy. In order to conserve energy, we must therefore say that the potential energy of these electrons has been converted to kinetic energy. Hence, the potential energy of the electron falls. A potential difference in a circuit merely means that work is done on electrons by an electric field as they pass through the circuit and their potential energy changes. Further, we can conclude that since the electric field is strongest in regions with high resistance, the potential difference must be greatest across these regions as well.
This might be a bit confusing since it implies that potential falls in the direction that electrons flow. Since we choose to deal with the flow of conventional current rather than electron flow, potential falls in the opposite direction of current flow.