Very good question!

Well, temperature of a gas is related to how fast the gas molecules are moving around, and how hard they crash into the walls of the container. Gas molecules move faster as the temperature of the gas increases, and when they move faster, they have more MOMENTUM, so the result is that they hit harder on the walls of the tires. Since the tires are made of rubber, they can expand - but only up to a certain limit.

So, the hotter you make the air in the tires, the faster the air molecules move around inside, and then they hit the walls of the tires harder. And since they are moving faster, they also crash into the tire walls more frequently (that is, more crashes per second), and the result is they push out the tires and make them expand. But, the rubber can only expand so far - so the pressure inside increases.

If you could heat up the air in a different container - one with walls that could expand without any resistance, then the pressure would not increase, if the volume of the container could increase as the temperature got hotter. But since there is a limit to the stretching of the tires, the pressure inside will increase as the air gets hotter. That is why you should fill your tires when they are cold, and leave a little room for expansion, because the air inside gets hotter when you drive.

Cold air decreases tire pressure because air atoms that are cold mover around less and need less space. If you ever go anywhere where it is really cold (below -15 degrees F) you will notice your car tires go flat on one side. And since the tire rubber is also cold and more stiff, if you move the car you will notice a bump every time you come around to the 'flat spot'.

As temperature increases, the pressure increases. If the tire did not expand or contract, (become larger or smaller) an increase in temperature will usually result in an increase in pressure.

Cold air is more dense. Meaning that there you can pack more molecules of oxygen into a smaller space.

If you have more molecules in a smaller space you have a greater pressure on that space. However, the air as a result becomes more compact. So, if you drive your car from the desert to the cold mountains and do not add any air to your tires you will get a decreased pressure in that space due to the molecules becoming more compact.