A person who gets angry when they... vs A person who gets angry when he
Solution 1:
They is used here just to be careful about the gender of the "person", as it is not clear whether the person is a male ("he") or female ("she").
Person is a generic word and could be both male or female.
Although, yes, "He" has been used as a default pronoun for all the genders, especially when one is not specific, but the nowadays, with people being extra sensitive about sexism and gender equality, it is preferred not to use a specific masculine pronoun.