I do not think there is any scientific explanation to why you get increased fuel efficiency with a full tank rather than with an almost empty one.
The issues involved here are the following:
A full tank has more fuel than an empty tank. That increases the mass of your car and as a result should actually reduce the mileage of your car. But when you compare the mass of the fuel in the car with the total weight of your automobile you will see that the mass of half a fuel tank of gasoline is negligible compared to the mass of your car. As a result the change in mass due to the fuel cannot explain the pretty large change in mileage of your car.
The second point to consider is the fact that fuel tanks in cars are designed to expel any fuel that evaporates. As the gas tank gets empty more of the empty space is filled with gas fumes and the result should be an increased loss in fuel. But the amount of fuel lost here, according to studies, is around 0.000952 gallon /year per tank which again is negligible. Also, modern cars are usually designed to direct the gas fumes back to the engine and there is practically no reason why this should affect mileage.
What could explain the difference is the fact that fuel tanks do not have a linear level sensor, i.e. the sensors used to detect the amount of fuel in the tank are non-linear in nature. When your fuel gauge shows that the tank is half full it actually is around two thirds empty. You unknowingly use more fuel by the time your fuel gauge drops from full to half empty. As there is around one third the fuel left when your gauge shows a half full tank, you would be able to travel a lesser distance with it. This can in a way explain the large difference in mileage. This explanation though is based on the assumption that your fuel level sensor is in reality a non-linear one.
Keeping the change in mileage based on fuel level aside, it would be much better to focus on things that have been scientifically proven to alter mileage like driving at an optimal speed, reducing the usage of your brakes, reducing the use of air conditioning, etc.