Motorists are told to check the pressure in their tires before going on a long trip for safety. Car tires are designed to operate within a specific pressure range as specified by the tire's manufacturer. The manufacturer determines this pressure based on the design and composition of the tire as well as from tests that they run. The reason that you check the pressure in the tire while the tires are cold (i.e. before you start driving the car) is because, when the tire heats up from friction while you drive, the air in the tire expands and actually increases the pressure in the tire. The pressure of a gas as a function of volume, amount, and temperature can be approximated by the ideal gas law: P*V=n*R*T.

In this equation P is the pressure of the gas, V is the volume that the gas is contained in, n is the amount of gas (usually measured in moles which is a measure of the number of atoms of the gas), R is called the ideal gas constant and is just a number (people typically use the units J/ (mol*K) so R in these units equals 8.314), and T is the absolute temperature (this is the temperature on a thermodynamic absolute scale meaning it starts at absolute zero which is 0 Kelvin). Using this equation, say your tire has a volume of 0.5 m³ (I am just making this number up for demonstration purposes. I do not know the actual volume of a tire), say it is about 70 degrees F outside (this is equal to about 294 K), and say your tire is at 25 psi (26psi ~ 179263.7 Pa).

From the ideal gas law: n=179263.7*0.5/ (8.314*294)= 36.7 mols.
If the manufacturer specified a pressure of 30 psi (206,843 Pa), and you look at the tire while it is hot (lets say 340 K), then you would see a pressure of P=36.7*8.314*340/0.5=207308.9 ~ 30 psi and would think that the tire is fine, when in fact the tire is under pressure.

This calculation was just for demonstration purposes and should not be used in actually filling up a tire. For that, always follow the directions specified by the tire's manufacturer. This demonstration; however, shows why you should check your tire pressure while it is at normal ambient temperature, and not while the tire is too hot (i.e. not after having been driven for a long period of time).

Tires are sealed, so the air inside them can't get out. Air exerts a certain amount of pressure on the container that confines it. Warm air exerts more pressure than cold air. Driving on roads produces friction, which heats up the tires and the air within. If the air inside heats up too much, the pressure can increase to where the tire will no longer hold it and the tire will pop - catastrophically.

Motorists have to check the pressure in car tires while the tires are cold because the air in the tire gets heated during driving, and since the volume of the tire is constant; that means that the tire pressure goes up as the air inside heats up.

Then you want to set the suggested pressure when the air in the tire is at same temperature as outside air, hence before you drive a bunch.

If we look at the ideal gas equation, PV = nRT, we can see that pressure is proportional to temperature. Now, while the air in our tires do not behave as "ideal gasses," they do still exhibit a proportionality between pressure in temperature. This means that when temperature decreases so does pressure. So on cold days, the pressure in our tires is lowered, and we should check to make sure that the pressure is at the "appropriate level." If we don't, we could compromise our safety in terms of the stability of the car, how well the car can turn corners, and how well the car will break. I hope this helps!