What is the opposite of a greeting? [duplicate]

Solution 1:

While it's not specific to ending a conversation, consider pleasantries:

An inconsequential remark made as part of a polite conversation

Solution 2:

These phrases are specifically called valedictions (WP) :

A valediction (derivation from Latin vale dicere, 'to say farewell'), or complimentary close in American English, is an expression used to say farewell, especially a word or phrase used to end a letter or message, or the act of saying parting words whether brief or extensive.

As to "why": to explicitly (and politely) inform you that the other party considers the conversation complete and ended: over and out.

Yours sincerely,

-Dan