Why is that when we take a shower, we don't get sick, but when we play in the rain, we get sick easily? Some sort of psychology, or does the rain contain some chemical substances?

Depends on the temperature of your shower and how long you stay in it. In northern climes, rain tends to be rather cold and if you're playing in it you're out for quite some time in wet clothing. A shower is usually warm to hot, you're in and out quick, and soon dried off. Lowering of body temperature means germs can work faster - which is why fevers and raised temperatures are part of the bodies reaction to infections as the raised temperature is not nice for the bugs.

It's not psychological, it's the water on your skin lowering your body temperature.
When you have a shower, you usually get dry straight away, and the shower is indoors so usually in a heated room. Therefore your skin temperature doesn't drop too far.
If you play in the rain, if it's cold the wet skin won't be able to maintain the body's optimum temperature as it's like sweating, as either way your wet and losing temperature.
The fact you tend to stay out in the rain with wet clothes and hair and therefore wet skin will reduce your core body temperature, and as a result will lower your resistance to the same germs that are wherever you get wet.