Why is the required tire pressure inversely proportional to the volume/size of the tire? I once noticed a tire pressure table on a farm recommended a tire pressure of around 1.0 bar (I don't remember the exact number, but it was definitely below 1.9) for the rear tire of farm tractors.
The recommended pressure on a car is around 2.2-2.5 bar.
My bike tire recommends 3.5 bar.
Why do smaller (slimmer? less voluminous?) tires require more pressure?
 A: The main considerations are grip and rolling efficiency.
A tyre dissipates energy as it flexes, and any energy dissipated in the tyre means extra effort from the rider or motor and therefore fewer miles per gallon. The more you pump up the tyre the harder it becomes and the less it flexes, so higher pressures are more fuel efficient.
However harder tyres also grip less well, so as you increase the pressure you reduce the grip e.g. in cornering and braking.
If you take the cyclist, here energy is at a premium so it's important that the tyres are as efficient as possible. Also grip is not usually a problem as the overall weights and therefore forces are relatively low. This makes a high pressure best.
Now consider a tractor. This is likely to be driving on slippery ground like muddy fields, so grip is essential. This makes it worth using a low tyre pressure and sacrificing some fuel economy.
The car is somewhere in the middle. Good fuel economy is nice, but then so is being able to brake hard. So the car uses tyre pressures somewhere in between the cycle and the tractor.
This is an oversimplification, because the design of cycle, car and tractor tyres is very different and the design affects the optimum pressure. However the description above captures the basic principles.
A: The principle is based on Laplace's law. Essentially, the wall or casing tension is tire firmness.  If this force (casing tension) value is held constant, increasing the volume (tire width) the inflation pressure goes down, conversely in a narrow road tire requires higher inflation pressure to achieve the casing or wall tension a larger tire would have at a much lower inflation pressure.
A: Needed pressure is all about real load/maximum load and pressure / pressure needed for maximum load.
but also speed is important.
The tire maker , calculates the maximum load for a reference pressure and reference speed, and prints this on the sidewall.
The goal of advice pressure for a load on tire is to give the tire the same deflection as when maximum load and reference pressure on tire. 
for lower speed the maximum load is higher then for higher speed.
I made a topic on American RV forum about it in wich I calculated back the system the tire makers use, so can be a little bit off the real system.
http://www.rv.net/forum/index.cfm/fuseaction/thread/tid/27477930.cfm
But Tractor tires use low speed so have higher maximum load then a truck tire with same sises would have.
That is why it allows low pressure .
But even car tires are lowered in the pressure when going off road , sometimes even as low as 0.8 bar for better gripp.
the advice for normal cars is kept high after 2000, before that year for smalller tires as low as 1.4 bar was sometimes given.
So because of the low speed and oversised tires a tractor has, it can do with lower pressure .  Also the agricultural ground is less pressed by the tractor so water is absorbed better.
A: If we treat the tire as completely flexible, the area of contact with the ground is the weight supported divided by the pressure.  A small tire will mean a small contact area. Your bicycle tire has a narrow contact patch and if the patch gets long the rim hits the ground, so you need reasonably high pressure.  My 25mm road bike tires run 6-9 bar.  
The tire also needs to be strong enough to contain the pressure.  The thickness of rubber will scale linearly with the pressure and also with the dimensions of the tire.  Making a car tire hold 6-9 bar would require extremely thick rubber, which would be so rigid it was uncomfortable and heavy.
