Here's why:
* Subject Pronoun: "He" is the subject pronoun, used when referring to the person speaking.
* Object Pronoun: "Him" is the object pronoun, used when the person is the object of the verb or preposition.
In the phrase "Answering your phone, this is he," "he" is the subject of the implied verb "am." We're saying "I am he" (the person you're calling).