Can anyone explain it in simple words? I tried to read about the Born rule on Wikipedia but it seems it is just describing the rule. Is there any proof that is suitable for person with no solid math background?
First, I would not call the Born rule a law as it is done in the Wikipedia article linked by OP in the question. It is, of course, the question of the definitions, but I would call the Born rule rather a postulate. A law in physics is a principle (usually universal) deduced from observations, while a postulate is an assumption, a statement that we can not prove or derive but which is assumed to be right and tested to be consistent with experimental observations.
The Born rule is a postulate, thus, there is no any proof of it; it is assumed to be right. The only thing we can do is to justify a postulate: we can make experimentally verifiable predictions on the basis of it and test them. And it happens so that so far the experimental outcomes are always in full agreement with the predictions done using the Born rule. Thus, we know it provides a correct description of physical systems.