发 (fā): This character means “to issue,” “to send out,” or “to express.” Picture it as an action of launching something forward, like issuing a command or expressing an idea.
言 (yán): This character means “speech,” “words,” or “to say.” The character originally depicted a mouth with a tongue, representing the act of speaking.
人 (rén): This character simply means “person” or “people.” It's a stylized drawing of a person walking.
When combined, 发言人 (fāyánrén) literally means “issue-speech-person,” a very logical construction for someone whose job is to issue official statements.