Proof of autentication is a means by which a campaign or airdrop asks you to create a post providing certain information which generally you are also asked to provide in a form upon signing-up to the campaign or airdrop.
For example, you may be asked to join a specific telegram group for a stake in a campaign, and then fill-in a form providing your telegram name, bitcointalk profile, bitcointalk name and ETH address. To prove that it was really you who provided the data, you are sometimes required to post a message in the campaign thread providing a required subset of the former provided information such as telegram name, bitcointalk profile and bitcointalk username in a given format.
Campaign manager will cross-reference the data before approving campaign stake rights.
I.e.
#Join-#Proof of authentication
Bitcointalk username: xxxxxxx
Bitcointalk profile link:
https://bitcointalk.org/index.php?action=profile;u=XxxxxxxxxTelegram profile link: @Xxxxx