Think of a conversation as three seats. The first seat belongs to whoever is speaking: I, or my group, we. The second seat belongs to the one being spoken to, and the third to everyone being spoken about. First person is the name of that first seat. English marks the seat clearly on pronouns and only on verbs, which is why 'I am' is the example everyone remembers.
From the word’s history
The term is a direct translation of Latin prima persona, which itself translated the Ancient Greek proton prosopon. Persona first meant a mask or a role in a play, so the grammar term keeps a faint picture of speaking as taking a part.
這個詞是拉丁文 prīma persōna 的直譯，而拉丁文那個說法又是翻自古希臘文 πρῶτον πρόσωπον。persona 最早指戲裡的面具或角色，所以這個文法術語裡還留著一點畫面：說話像上台，第一個角色就是說話的人自己。
Wiktionary entry 'first person' (kaikki extract supplied in packet): 'Calque of Latin prīma persōna, itself a calque of Ancient Greek πρῶτον πρόσωπον.'