The three pronoun format (e.g., "he/him/his") is an obvious and hard-to-misunderstand way of communicating a person's personal gender pronouns. It is just convention, and even then, the two pronoun format (e.g., "he/him", dropping the possessive) is even more popular now. Just "he" wouldn't be very obvious, and could even be mistaken for "she" when spoken. And, as an aside, I've never seen more than 3 forms of the same pronoun listed either, since that would just increase the amount of words used without making the message any more clear.
The reason personal gender pronouns are even stated at all is primarily fueled by the LGBTQ community, which has many members who are gender nonconforming and whose pronouns cannot be (easily) guessed. However, stating pronouns also benefits people outside the LGBTQ community, such as anyone with a gender-neutral name or appearance.
According to an NPR interview (no relation to me):
Why are pronouns typically given in the format "she/her" or "they/them" rather than just "she" or "they"?
The different iterations reflect that pronouns change based on how they're used in a sentence. And the "he/him" format is actually shorter than the previously common "he/him/his" format.
"People used to say all three and then it got down to two," Heng-Lehtinen [deputy executive director of the National Center for Transgender Equality] laughs. He says staff at his organization was recently wondering if the custom will eventually shorten to just one pronoun. "There's no real rule about it. It's absolutely just been habit," he says.
But he notes a benefit of using he/him and she/her: He and she rhyme. "If somebody just says he or she, I could very easily mishear that and then still get it wrong."
The interview also explains what it means when there are two different pronouns listed, like "she/they": it's used to indicate the person wants to be referred to with either/both pronoun sets. (In the case of "she/they", that's "she/her" and "they/them".) While "he/her/them" (mixing cases of pronouns) would not be used, some people do go by the pronouns "he/she/they", like one person in the article How To Affirm the People in Your Life Who Use Multiple Sets of Pronouns. Another rare occurrence is people who go by "any pronouns", which is similar to "he/she/they" but technically includes neopronouns.
See also my answer to Is it correct to say your pronouns are "she/they" or should it be "she/them"?.