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The lower the temperature, the lower the pressure inside of the sphere. If the exact values were plotted out, a linear relationship would be apparent. Extrapolating this line to the point where there would be no pressure yields absolute zero, which is about -273.15 degrees Celsius.
Pressure is caused by the collisions of gas particles with each other and whatever objects they may collide with. When the temperature is lowered, the particles move more slowly, decreasing the frequency and strength of these collisions. In turn, the pressure falls.
Absolute zero can be defined as the temperature at which matter does not move. At absolute zero, even subatomic vibrations are put to a grinding halt. Because the pressure in this experiment is caused by the movement of a gas, the pressure would cease to exist when the gas stops moving (a.k.a. absolute zero). Therefore, when the linear relationship discovered in this experiment is extrapolated to the point where the pressure is zero, the corresponding temperature is absolute zero.