Why does low tire pressure increase friction?

Lower pressure increases surface contact and increases static friction, and static friction does not involve heat loss, so that is good. But rolling friction is not good and does involve heat loss.

Rolling friction heating is due to the inelastic deformation the rubber of the tire experiences when it is in contact with the road. See this article on rolling resistance from Wikipedia: https://en.wikipedia.org/wiki/Rolling_resistance

When the rubber is in contact with the road for each revolution, it is compressed, it then expands when it leaves the surface. The compression and expansion is not perfectly elastic, thus there is heat loss in the form of friction. The lower the tire pressure, the more rubber that is in contact with the road for each revolution, and the greater the friction heat loss. These increased heat losses add up to lower fuel economy.

Hope this helps.

Static friction does not actually reduce your fuel economy. Without it, your wheels would slip, so it is a good thing.

What causes the loss of fuel economy is rolling resistance. This is due to hysteresis; energy lost due to the cyclic deformation of the tyre as it rolls.

It's not a great analogy but imagine the partially inflated tire as a disc with the axel off centre. When rolled the axel will oscillate up and down - i.e. some energy is lost from being able to contribute to the forward motion.

In your case the axel doesn't bob up and down but energy is lost to reshaping the tite from bulging on one side to bulging on the other with each half revolution.

When using pneumatic tires, the "rolling friction" represents the effort required to continuously reshape the tire. The lower the pressure in the tire, the greater the amount of deformation as it rolls, and thus the more work will be required to achieve such deformation. If one were to inflate tires sufficiently that they would barely deform as they rolled, this friction would be greatly reduced (thus improving fuel economy) but the performance and handling advantages of pneumatic tires would be sacrificed.

If one were driving a car with absolutely rigid wheels, then every little bump in the road surface would cause the force between the wheel and the ground to be increase as the car hits the bump, and then decrease as soon as the wheel is over. Because stopping force is limited by the breaks as well as traction, momentary increases in traction won't improve it, but momentary decreases in traction will reduce it. Pneumatic tires minimize this problem because the contact area can be continuously reshaped to minimize the increase in force when hitting a bump, and thus the inevitable reduction in force afterward.

The reason that tire pressures are stated on vehicles rather than tires is that the amount of pressure needed to limit deformation to the optimal level will vary depending upon a vehicle's weight. Ideally, tires would supply a table of recommended inflation pressures based upon vehicle weight, but most vehicles are intended for use with a relatively narrow range of tire sizes, and tires of a given size will tend to have similar inflation recommendations. The exact optimal pressure for different manufacturers' tires won't be identical, but most tires will perform reasonably well over a sufficiently large range of pressures that a pressure chosen for a typical tire will usually work reasonably well--even if not quite optimally--for others tire of similar size.