Thermal conductivity has dimensions of power/(length * temperature) Power is the rate of heat flow, i.e. energy flow in a given time.  Length represents the thickness of the material the heat is flowing through, and temperature is the difference in temperature through which the heat is flowing.

In SI units, it is commonly expressed as Watts/meter/K, and in US units it is commonly given in BTU/hr/ft/degree F.

It expresses the rate at which heat is conducted through a unit thickness of a particular medium.  That rate will vary linearly based on the temperature difference across the material, so it is expressed as a value per degree of temperature difference, thus Heat Rate per unit thickness per degree of temperature difference.