They mean state as in state of matter (solid, liquid, gas, etc.). Because it takes the release/absorption of energy to transition between states of matter, the internal energy of a substance depends on its state of matter. For example, water vapor has a higher internal energy than the same quantity of liquid water even if they are at the same temperature, pressure, etc.
The difference in internal energy comes from the fact that different states of matter are associated with different intermolecular bonds that make different contributions to the substance's chemical potential energy.