Well, I don't know if this is as specific as you asking, but the pressure in the tire has to balance the external forces acting on the tire. In this situation, there are two to consider: Air pressure pushing in on the tire and the amount the earth pushes up on the tire (3rd law equivalent of the weight of the car).
When the car is up on a lift, the only pressure it feels is the air pressure pushing in. The air pressure in the tire compensates to balance this by changing volume (Ideal gas law). This pressure at sea level is roughly 14 psi and is dependent on the entire surface area of the tire.
When the car is dropped down to the ground, you now have the equivalent of the weight of the car distributed in each of the tires in addition to the air pressure there before (this is slightly smaller since part of the surface area is not in contact with air). So in this case the pressure is related to both the contact area and the total surface area.
A simpler approach is to look at the tire. When up on the lift they have a set volume. When dropped to the ground, they are compressed. As the volume goes down (with fixed temperature) the pressure has to go up.