This is a commonly used idiom: he has a chip on his shoulder. Some people use it to indicate that he is behaving arrogantly, normally due to his success or position. Perhaps they are making a connection between the chip on his shoulder and a microprocessor chip or a blue chip stock. Actually, this is not the meaning of the idiom. It is not related to being arrogant.
When someone has a chip on his shoulder, he is angry or feels disrespected. And he feels angry/disrespected because he is under the impression that he has been treated unfairly in the past.
And that is what it means to have a chip on one’s shoulder.
A venture capitalist once said that he looks for an entrepreneur who has a chip on her shoulder, as this means that she will try even harder to succeed. Does that make sense?