You can turn the question around: why do skinny tires require more pressure? The main answer is, to prevent pinch flats. Suppose you drop off a curb. You don't want the rim to hit the ground. The fat tire has two advantages: 1) because it is fatter, it can put more rubber on the road as the rim moves toward the road. So that give more area over which the pressure can act; 2) because it is fatter, it has more distance over which to act before the rim hits. So that would be one criterion for sufficient pressure: enough to avoid a pinch flat when dropping off e.g. a 9 inch height. A skinny tire just requires more pressure.

Another aspect of the puzzle: fat tires cannot handle as much pressure as skinny tires can. There are two problems. The pressure can pull the tire apart, or the pressure can pull the rim apart. One important additional variable in all this is the rim width. A fat tire on a narrow rim will pull the rim sides apart more. The same tire on a wider rim will have the tire exiting the rim at a less horizontal direction and so exert less spreading force on the rim.

One way to think about the effect of pressure on the tire itself is to think about a clothes line. If the clothes line is not exactly taut but still has very little slack, so it runs almost straight between its supports, then when you hang a wet towel in the middle, the tension in the line will get quite high. If the line isn't very strong, it could snap. If the line has quite a bit of slack, then the tension won't be so high.

At the limit, where the line is very long, the tension will be just half the weight of the wet towel. On each side of the towel the line just pulls half the weight of the towel upwards. But if the line started quite straight... actually the tension can be extremely high. Each side still has to pull up with half the weight, but because the line is almost horizontal only a small part of the tension is upward. Most of the tension will be horizontal, the tension pulling away from the towel in either direction.

A wide tire doesn't curve sharply. That shallow curve still has to push in to hold the air pressure, but the tension mostly pulls away, stretching the tire surface. With a narrow tire that sharper curve means that more of the casing tension goes to pushing against the air pressure.

Here is my nifty tire pressure table: Interdependent Science: Bicycle Tire Pressure